Analysis and Applied Mathematics Seminar : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Sept. 14, 2015
Wolfgang Gaim :
4 p.m. in SEO 636
Abstract
In his 1932 paper, Eugene Wigner introduced the now famous Wigner function in order to compute quantum corrections to classical equilibrium distributions. We show how to extend this program and compute semiclassical approximations to time evolved quantum observables as well as quantum mechanical equilibrium distributions for slow, semiclassical degrees of freedom coupled to fast, quantum mechanical degrees of freedom. The main examples are molecules and electrons in crystalline solids. The semiclassical formulas contain, in addition to quantum corrections similar to those of Wigner, also modifications of the classical Hamiltonian system used in the approximation: The classical energy and the Liouville measure on classical phase space turn out to have non-trivial-expansions in the semiclassical parameter. This talk is based on joint work with Stefan Teufel.
Sept. 21, 2015
George Nenciu :
4 p.m. in SEO 636
Abstract
Basic facts of Kato-Rellich regular perturbation for discrete eigenvalues are
briefly reviewed. For singular perturbations (e.g. anharmonic oscillator,
Stark effect) asymptotic expansions for the perturbed projections leading to
(almost) invariant subspaces are provided.
This is a two hour introductory talk intended mainly for graduate students.
Sept. 28, 2015
George Nenciu :
4 p.m. in SEO 636
Abstract
Perturbation theory for embedded eigenvalues and exponential decay for resulting metastable states are considered.
The relations between the formal Rayleigh-Schrödinger expansion, spectral concentration and decay law for metastable states are discussed in the smooth setting.
The main result is that if the FGR constant vanishes then the first order correction in the Rayleigh-Schrödinger expansion
is well defined and the exponential decay law for the corresponding metastable state has both the decay rate and error term of order $\epsilon^4$
where $\epsilon$ is the perturbation strength.
Oct. 5, 2015
Fred Hickernell :
4 p.m. in SEO 636
Abstract
Popular adaptive numerical algorithms aim to return an answer within the user’s error tolerance with an appropriate amount of computational effort—harder problems require more effort and easier problems require less effort. The error estimates assumed by these adaptive algorithms often lack theoretical justification. This talk describes some of the pitfalls of these algorithms and our recent efforts to provide adaptive algorithms with rigorous guarantees. We focus on integration, interpolation and optimization problems.
Oct. 12, 2015
Leonid Berlyand :
4 p.m. in SEO 636
Abstract
We study the motion of a eukaryotic cell on a substrate and investigate the dependence of this motion on key physical parameters such as strength of protrusion by actin filaments and adhesion. This motion is modeled by a system of two PDEs consisting of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network.
The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion perturbed by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters. In the subcritical regime, the well-posedness of the problem is proved (M. Mizuhara et al., 2015). Our main focus is the supercritical regime where we established surprising features of the motion of the interface such as discontinuities of velocities and hysteresis in the 1D model, and instability of the circular shape and rise of asymmetry in the 2D model.
Because of properties (i)-(ii), classical comparison principle techniques do not apply to this system. Furthermore, the system can not be written in a form of gradient flow, which is why Γ-convergence techniques also can not be used. This is joint work with V. Rybalko and M. Potomkin.
Oct. 19, 2015
Chang-Yeol Jung :
4 p.m. in SEO 636
Abstract
A semi-analytical method is developed based on conventional integrating factor (IF) and exponential time differencing (ETD) schemes for stiff problems.
The latter means that there exists a thin layer with a large variation in their solutions. The occurrence of this stiff layer is due to the multiplication of a very small parameter $\epsilon$
with the transient term of the equation. Via singular perturbation analysis, an analytic approximation of the stiff layer, which is called a corrector, is sought for and embedded into the IF and ETD methods. These new schemes are then used to approximate the non-stiff part of the solution. Since the stiff part is resolved analytically by the corrector, the new method outperforms the conventional ones in terms of accuracy. In this paper, we apply our new method for both problems of ordinary differential equations and some partial differential equations.
Nov. 2, 2015
Youngjoon Hong :
4 p.m. in SEO 636
Abstract
In this talk, I will present convergence results of singularly perturbed problems in the sense of PDEs, which is related to the vanishing viscosity limit. I also provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a P1 classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh.
Nov. 9, 2015
Bing-Yu Zhang :
4 p.m. in SEO 636
Abstract
In the past three decades, harmonic analysis has played important roles in the rapid advances of the study of nonlinear dispersive wave equations. In particular, many new tools have been developed to establish various well-posedness results for the pure initial value problems of nonlinear dispersive wave equations. However, how those harmonic analysis based tools can be used effectively to study non-homogeneous boundary value problems of nonlinear dispersive wave equations is still yet to be investigated. In this talk, we will introduce the concept of the boundary integral operators and show how they play important roles in studying non-homogeneous bound- ary value problems of nonlinear dispersive wave equations. In particular, we will demonstrate through examples of the KdV equation and the Schr ̈odinger equation how the boundary integral operators can enable us to use those harmonic analysis based tools to study non-homogeneous boundary value problems effectively.
Nov. 16, 2015
Brian Seguin :
4 p.m. in SEO 636
Abstract
I will present a microscopic model for plant cell wall biomechanics that takes into account both the microstructure coming from the cellulose microfibrils and the chemical reactions between the cell wall’s constituents. Particular attention is paid to the role of pectin and the impact of calcium-pectin cross-linking chemistry on the mechanical properties of the cell wall. An outline of how to prove existence and uniqueness of the microscopic system will be given. Finally, a multiscale analysis in the form of homogenization is carried out to obtain a macroscopic model. I will include a brief introduction to homogenization in my talk.
Nov. 23, 2015
Karen Zaya :
4 p.m. in SEO 636
Abstract
Fundamental mathematical questions about the 3D Navier-Stokes remain unanswered, such as the question of the regularity of solutions to the equations. Thus it is natural to ask: If we assume a smooth solution to the 3D Navier-Stokes equations $u$ loses regularity at time $T^*$, what is the rate of blow-up? In this talk, we discuss blow-up rates of solutions in the homogeneous Sobolev spaces, in particular the new result in $\dot{H}^\frac{3}{2}$. We will also discuss a newly developed regularity criterion for the 3D Boussinesq equations, which only imposes a condition on the low modes of the velocity $u$. The key tool in the development of this weaker regularity criterion is linked to the dissipation wave number.
Jan. 25, 2016
Hao Jia :
4 p.m. in SEO 636
Abstract
In this talk, I will firstly give an overview of significant developments in the study of focusing energy critical wave equations, with emphasis on characterizing Type II, i.e., bounded solutions. The central question considered will be the soliton resolution conjecture which predicts that all Type II solutions will asymptotically decouple into a sum of traveling waves, plus a radiation term in the global case or a regular term in the finite blow up case. A very recent proof of this conjecture for a sequence of times will be sketched. The proof is based on joint works with Duyckaerts, Kenig and Merle.
Feb. 1, 2016
Christian Ringhofer :
4 p.m. in SEO 636
Abstract
In this talk we present a kinetic framework for the time evolution of multi agent systems where individual agents make decisions based on concepts of behavioral and evolutionary game theory. As an application, we design optimal
strategies for the design of insurance policies.
Feb. 8, 2016
Brittany Froese :
4 p.m. in SEO 636
Abstract
The problem of optimal transportation, which involves finding the most cost-efficient mapping between two measures, arises in many different applications. However, the numerical solution of this problem remains extremely challenging. We describe a numerical method for the widely-studied case when the cost is quadratic. The solution is obtained by solving the Monge-Ampere equation, a fully nonlinear elliptic partial differential equation (PDE), coupled to a non-standard implicit boundary condition. Expressing this problem in terms of weak (viscosity) solutions enables us to construct a monotone finite difference approximation that computes the correct solution. A range of challenging computational examples demonstrate the effectiveness of this method, including the recent application of this method to problems in beam shaping and seismic inversion.
Feb. 15, 2016
Songsong Lu :
4 p.m. in SEO 636
Abstract
I will review some resent results on the long-time behavior of the nonautonomous 3D Navier-Stokes equations and general nonautonomous reaction-diffusion systems. The method is based on a new framework of evolutionary systems that deals directly with the notion of a uniform global attractor due to Haraux, and with which a trajectory attractor can be defined for the original system under consideration. The notion of a trajectory attractor was previously established for a system without uniqueness by considering a family of auxiliary systems including the original one. I will also expound on the existence of a strongly compact strong trajectory attractor when the system is asymptotically compact, and how we view the global and trajectory attractors in a unified way. Part of the results of the talk is a joint work with Cheskidov.
Feb. 22, 2016
Andrew Sward :
4 p.m. in SEO 636
Feb. 29, 2016
Mahmood Jokar :
4 p.m. in SEO 636
March 7, 2016
Romeo Awi :
4 p.m. in SEO 636
Abstract
This talk is concerned with some numerical aspects of Morrey's Conjecture in dimension
2x2. The problem is to know whether there exists a rank-one convex
function defined on 2x2 matrices that is not quasiconvex. The conjecture
is first cast as a minimization problem. The later problem is
approximated by minimization problems over families of affine piecewise
functions. Two methods that could lead to finding counter-examples are
suggested.
March 14, 2016
Shuwen Lou :
4 p.m. in SEO 636
Abstract
We will introduce fractal properties of rough differential equations driven by frational Brownian motion with Hurst parameter H>1/4. We will first survey some known results on density and tail estimates of such processes. Then we will show the Hausdorff dimension of the sample paths is equal to min(d, 1/H), where d is the dimension of the process. Also we will show that with positive probability, the level sets in the form of {t: X_t=x } has Hausdorff dimension 1-dH when dH<1, and are almost surely empty otherwise.
March 28, 2016
Gustavo Ponce :
4 p.m. in SEO 636
Abstract
In a joint work with P. Isaza and F. Linares we show that solutions of the IVP
for the $k$-generalized KdV equation
\begin{equation}
\begin{cases}
\begin{aligned}
\label{aaa}
&\partial_t u + \partial_x^3 u +u^k\partial_x u=0,\;\;\;\;\;t,\;x\in\mathbb R,\;\;k\in\mathbb Z^+,\\
&u(x,0)=u_0(x)
\end{aligned}
\end{cases}
\end{equation}
preserve some smoothness of the initial data $u_0$ and that this regularity moves with infinite speed to its left as time evolves.
April 4, 2016
David Ambrose :
4 p.m. in SEO 636
Abstract
In this talk, we will discuss the initial value problem for 3D interfacial fluid flow with surface tension.
We will emphasize the case in which the fluid velocities are given by Darcy's Law; this can serve as a
model for intefacial flow in a porous medium. We will discuss a well-posedness proof for the problem,
with the initial data in Sobolev spaces. We will also discuss a non-stiff numerical method; this
is similar in spirit to the method of Hou, Lowengrub, and Shelley for the corresponding 2D problem.
Finally, we will give some of the details of a convergence proof for a variant of this numerical method;
this convergence proof requires estimates which are similar in spirit to the estimates required to
demonstrate well-posedness. While convergence proofs for boundary integral methods had been
developed previously for 2D flows with or without surface tension or for 3D flows without surface
tension, this is the first convergence result for a boundary integral method for a 3D flow with
surface tension. The results discussed include joint work with Yang Liu, Nader Masmoudi, Michael
Siegel, and Svetlana Tlupova.
April 11, 2016
Burak Erdogan :
4 p.m. in SEO 636
Abstract
In this talk, we will discuss recent results on a smoothing property of nonlinear dispersive PDE stating that the nonlinear part of
the evolution is smoother than the initial data, and some applications of this phenomenon. We will concentrate on the cubic NLS equation on the
torus, the real line, and the half-line.
April 18, 2016
Eduard Kirr :
4 p.m. in SEO 636
Abstract
Hamiltonian PDE's model a vast array of wave phenomena and,
consequently, exhibit special solutions, called coherent structures, among
which the solitary waves (solitons) are the best known examples. Despite
great differences in the underlying physical models similar mathematical
techniques are used to study the existence of coherent structures and
their influence on the evolution of general solutions. I will review these
methods and present a new approach capable of finding all coherent
structures supported by a given Hamiltonian PDE and their stability
properties.
April 25, 2016
Sara Pollock :
4 p.m. in SEO 636
Abstract
I will introduce a class of nonlinear elliptic problems featuring solution-dependent and gradient-dependent diffusion. I will discuss some of the ways standard solution techniques can fail to resolve these nonlinear problems when they feature thin layers and steep gradients in their coefficients; and, I will introduce a framework that can be used to solve such problems starting from a coarse finite element discretization. The framework features a sequence of partial solves of regularized problems used to determine refinement of the mesh to resolve the problem coefficients and data. Adaptivity is used both for mesh refinement and automatic control of the regularization parameters to ultimately solve the discrete problem efficiently and without regularization. Numerical examples will illustrate the ideas and demonstrate the presented algorithm.
Aug. 29, 2016
Jiahong Wu :
4 p.m. in SEO 636
Abstract
The Boussinesq equations concerned here model geophysical flows such as atmospheric fronts and ocean circulations. In addition, they play an important role in the study of Rayleigh-Benard convection. Mathematically the 2D Boussinesq equations serve as a lower-dimensional model of the 3D hydrodynamics equations. The global regularity problem on the 2D Boussinesq equations with partial or fractional dissipation has attracted considerable attention in the last few years. This talk presents some recent work on the 2D Boussinesq equations with general critical dissipation as well as the global regularity result on the 2D Boussinesq equations with vertical dissipation. If time permits, we will also briefly discuss the regularity problem on the partially dissipated Boussinesq equations in a bounded domain.
Sept. 12, 2016
Peijun Li :
4 p.m. in SEO 636
Abstract
This talk concerns the source scattering problems for acoustic wave propagation, which is governed by the two- or three-dimensional stochastic Helmholtz equation. As a source, the electric current density is assumed to be a random function driven by an additive colored noise. Given the random source, the direct problem is to determine the radiated random wave field. The inverse problem is to reconstruct statistical properties of the source from the boundary measurement of the radiated random wave field. In this work, we consider both the direct and inverse problems. We show that the direct problem has a unique mild solution via a constructive proof. Using the mild solution, we derive effective Fredholm integral equations for the inverse problem. A regularized Kaczmarz method is developed by adopting multi-frequency scattering data to overcome the challenges of solving the ill-posed and large scale integral equations. Numerical experiments will be shown to demonstrate the efficiency of the proposed method. The framework and methodology developed here are expected to be applicable to a wide range of stochastic inverse source problems.
Sept. 19, 2016
Robert Buckingham :
4 p.m. in SEO 636
Abstract
The three-wave resonant interaction equations describe the time evolution of the complex amplitudes of three resonant wave modes. We analyze the collision of two or three packets in the semiclassical limit by applying the inverse-scattering transform. Using WKB analysis, we construct an associated semiclassical soliton ensemble, a family of reflectionless solutions intended to accurately approximate the initial data in the semiclassical limit. Plots of the soliton ensembles indicate the space-time plane is partitioned into regions containing either quiescent, slowly varying, or rapidly oscillating waves. This behavior resembles the well-known generation of dispersive shock waves in equations such as the Korteweg-de Vries and nonlinear Schrodinger equations, although the physical mechanism must be different as the system is non-dispersive. This is joint work with Robert Jenkins and Peter Miller.
Oct. 10, 2016
Daniela Valdez-Jasso :
4 p.m. in SEO 636
Abstract
TBA
Oct. 17, 2016
Robert Eisenberg :
4 p.m. in SEO 636
Abstract
Life is different because it is inherited. All life comes from a blueprint (genes) that can only make proteins. Proteins are studied by more than 10^5 scientists and physicians every day because proteins are so important in health and disease. The structure of proteins is so important that governments have spent billions of dollars measuring them in atomic detail. But the forces that govern the movement and function of proteins are not visible in the structure. Mathematics is needed to compute both function and forces so comparison with experiment can be made. An important class of proteins—ion channels—have been analyzed successfully with mathematics. A consistent mathematical description produces macroscopic features of the atomic detailed structures that fit data in a wide range of conditions surprisingly well, using only a handful of parameters, never changed. I will present the Poisson Nernst Planck approach to the open ionic channel (and its improvements), trying to understand the stochastic meaning of the PNP equations, even trying to ‘derive’ them. PNP describes the mean field correlation of point charges, but the particles of biology are nothing like points. Extensions of PNP that deal with many of the resulting correlations will be discussed. Systems of crowded charge like those found in biology are so crowded that ‘everything interacts with everything else’. The result is a complex fluid. I will present the energetic variational approach to such systems, developed by Chun Liu, more than anyone else.
Oct. 24, 2016
Ming Chen :
4 p.m. in SEO 636
Abstract
We consider 2D steady water waves with heterogeneous density. The presence
of stratification allows for a wide variety of traveling waves, including
fronts, so-called generalized solitary waves with ripples in the far
field, and even fronts with ripples! Among these many possible wave
patterns, we prove that for any smooth choice of upstream velocity and
monotone streamline density function, there always exists a continuous
curve of solitary waves with large amplitude, which are even and
decreasing monotonically on either side of a central crest. As one moves
along this curve, the horizontal fluid velocity comes arbitrarily close to
the wave speed.
We will also discuss a number of results characterizing the qualitative
features of solitary stratified waves. In part, these include bounds on
the Froude number from above and below that are new even for constant
density flow; an a priori bound on the velocity field and lower bound on
the pressure; a proof of the nonexistence of monotone bores for stratified
surface waves; and a theorem ensuring that all supercritical solitary
waves of elevation have an axis of even symmetry. This is a joint work
with Samuel Walsh and Miles Wheeler.
Oct. 31, 2016
Colette Guillopé :
4 p.m. in SEO 636
Abstract
This is joint work with Jerry Bona (UIC) and Thierry Colin (Université de Bordeaux).
This talk is concerned with long-crested waves such as those arising in bore propagation. Such motions take place on rivers when a surge of water invades an otherwise quiescent stretch and in the run-up of waves in the near-shore zone of large bodies of water.
In an earlier work, we developed an idealized model for such waves based on a Boussinesq system of equations. The local well-posedness theory developed in this earlier work applies to the sort of initial data arising in modeling bore propagation.
In this talk we shall deal more specifically with well posedness on the longer, Boussinesq time scale. Without a well-posedness theory at least on the Boussinesq time scale, the model may not be of any practical use. The issue of well-posedness is complicated by the fact that the total energy of the idealized initial data is infinite.
Nov. 7, 2016
Kui Ren :
4 p.m. in SEO 636
Abstract
Hybrid imaging (also called multiwave imaging) refers to imaging techniques where we combine a high-resolution imaging method (such as ultrasound imaging) with a high-contrast imaging method (such as diffuse optical tomography) to take the advantages of both imaging modalities. Imaging reconstructions in hybrid imaging usually involve solving inverse problems for PDEs of different types. In this talk, I will provide a quick overview of the field of hybrid imaging and demonstrate, by looking into the details of two hybrid modalities, many challenges and opportunities in this fascinating field for applied mathematicians.
Nov. 14, 2016
Luis Silvestre :
4 p.m. in SEO 636
Abstract
The Boltzmann equation describes the evolution of particle densities, in terms of space and velocity, for gases and plasma. We will discuss the regularization effect of this equation in the non cutoff case. Our analysis is based on techniques that originate in the study of parabolic integro differential equations.
Nov. 21, 2016
Youngjoon Hong :
4 p.m. in SEO 1227
Abstract
Numerical methods for the solution of initial value problems in partial differential equations have made enormous progress over the past decades. The needs in mathematical modelling of efficient numerical algorithms as an alternative to classical methods of applied mathematics have remarkably increased. In this respect, there are many of developments of numerical methods with aid of analytical methods to understand the mechanism and physical/mathematical structures of model problems.
I will present recent developments of numerical methods and computations for physical models emerging from geophysics, fluid dynamics, and electromagnetism with aid of analysis. More precisely, we mainly focus on the numerical methods for geophysical fluid model (primitive equations), scattering problem in layered media, and singularly perturbed problems.
Jan. 30, 2017
Jinxin Xue :
4 p.m. in SEO 636
Abstract
It is an important open problem in dynamical systems to prove positive Lyapunov exponent for the standard map. In this talk I will introduce the problem, explain its significance and discuss the main difficulty. I will also talk about my joint work with Blumenthal and Young on this problem: by introducing a extremely small random perturbation to the map, we prove large positive Lyapunov exponent for the randomized system.
Feb. 20, 2017
Andrei Tarfulea :
4 p.m. in SEO 636
Abstract
We consider a model for three-dimensional fluid flow on the
torus that also keeps track of the local temperature. The momentum
equation is the same as for Navier-Stokes, however the kinematic viscosity
grows as a function of the local temperature. The temperature is, in turn,
fed by the local dissipation of kinetic energy. Intuitively, this leads to
a mechanism whereby turbulent regions increase their local viscosity and
dissipate faster. We prove a strong a priori bound (that would fall within
the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on
the thermally weighted enstrophy for classical solutions to the coupled
system.
Feb. 27, 2017
Stanley Snelson :
4 p.m. in SEO 636
Abstract
We consider the Landau equation, an integro-differential kinetic model from plasma physics that describes the evolution of a particle density in phase space. It arises as the limit of the Boltzmann equation when grazing collisions predominate. I will give an overview of prior work on the regularity theory of the Landau equation, and describe how to prove a priori upper bounds that decay polynomially in the velocity variable. The technical tools include precise bounds on the coefficients, and tracking how local estimates scale as the velocity grows. I will also explain why the polynomial decay cannot be improved to exponential decay. This talk is based on joint work with Stephen Cameron and Luis Silvestre.
March 6, 2017
Jun Kitagawa :
4 p.m. in SEO 636
Abstract
We consider a damped Newton algorithm to calculate the transport map of semi-discrete optimal transport problems, where the initial measure is absolutely continuous while the target measure is discrete. For costs satisfying standard conditions arising in the regularity theory of optimal transport, and initial measures with Hölder continuous density and support satisfying some mild connectivity conditions, we show this algorithm enjoys global linear and local superlinear convergence. A key ingredient in the proof comes from PDE theory, and involves Loeper’s geometric interpretation of the Ma-Trudinger-Wang conditions. This is joint work with Quentin Mérigot and Boris Thibert.
March 13, 2017
Jack Arbunich :
4 p.m. in SEO 636
Abstract
We present recent work (jointly with C. Sparber) on the weakly nonlinear wave dynamics occurring in cubic nonlinear Schrödinger equations with a periodic honeycomb lattice potential. Using a semiclassical scaling, we rigorously derive an effective macroscopic model of nonlinear Dirac type. To this end we employ a multi-scale asymptotic expansion together with rigorous error estimates in certain scaled Sobolev spaces. If time permits, we shall also discuss the case of a non-local Hartree nonlinearity.
March 27, 2017
Mark Groves :
3 p.m. in SEO 612
Zhiwu Lin :
4 p.m. in SEO 636
Abstract
Consider
generalized KDV equations with a power non-linearity (u^p)_x. These KDV
equations have solitary traveling waves, which are linearly unstable
when p>5 (supercritical case). Jointly with Jiayin Jin and Chongchun
Zeng, we constructed invariant manifolds (stable, unstable and center)
near the orbit of the unstable traveling waves in the energy space. In
particular, the local uniqueness and orbital stability of the center
manifold is obtained. These invariant manifolds give a complete
description of the dynamics near unstable traveling waves.
April 3, 2017
Trevor Leslie :
4 p.m. in SEO 636
Abstract
When a Leray-Hopf weak solution to the Navier-Stokes equations has a singularity set $S$ of dimension $d$ less than 3--for example, a suitable weak solution--we find a family of $L^q L^p$ conditions that guarantee validity of the energy balance relation. Our conditions surpass the classical Lions-Ladyzhenskaya $L^4 L^4$ result in the case $d<1$. In this talk, we focus on the special case when $S$ belongs to a single time-slice. Besides allowing more flexibility in the relevant analysis (and accordingly, stronger results), the time-slice case is the one which is most relevant for the blowup problem. If time allows, we will also discuss extensions to the fractional Navier-Stokes equations.
April 10, 2017
Juhi Jang :
4 p.m. in SEO 636
Abstract
I will review stability problems of Lane-Emden star configurations modeled by the Euler-Poisson system and present a recent joint work with Mahir Hadzic on the global-in-time existence of expanding stars in the mass-critical regime.
April 12, 2017
Romeo Awi :
2 p.m. in SEO 1227
Abstract
In elasticity theory, the energy functionals are convex functions of
the minors of the deformation gradient and thus are not expected to be
convex. In this case, the uniqueness of equilibrium configurations is a
challenging topic and has been linked to the fact that the potential
map transports Lebesgue measure into a measure absolutely continuous
with respect to Lebesgue measure. We will discuss limit cases where the
potential may be degenerate and we still have uniqueness under mild
regularity conditions on the maximizers of the dual problem.
April 17, 2017
Cecilia Mondaini :
4 p.m. in SEO 636
Abstract
The purpose of this talk is to present some analysis results concerning a feedback-control (nudging) approach for data assimilation that works for a general class of dissipative dynamical systems and observables. First, I will consider the situation when the measurements are discrete in time and contaminated by systematic errors. In this case, we obtain an estimate for the error between the approximating solution and the reference solution that shows exponential convergence in time modulo the bound on the errors. Later, I will consider a numerical approximation of the nudging equation via the Postprocessing Galerkin Method, and show an analytical estimate of the truncation error committed in this finite-dimensional approximation. Most importantly, this error estimate is uniform in time. This is in contrast with the error estimate for the usual Galerkin approximation of the 2D Navier-Stokes equations, which grows exponentially in time. This talk is based on joint works with C. Foias and E. S. Titi.
April 24, 2017
Zoran Grujic :
4 p.m. in SEO 636
Abstract
It is shown--within a mathematical framework based on the suitably
defined scale of sparseness of the super-level sets of the positive
and negative parts of the vorticity components, and in the context of
a blow-up-type argument--that the ever-resisting "scaling gap" in the
3D Navier-Stokes regularity problem can be reduced by an algebraic
factor; all preexisting improvements have been logarithmic in nature,
regardless of the functional set up utilized. The mathematics
presented was inspired by morphology of the regions of intense
vorticity/velocity gradients observed in computational simulations of
turbulent flows. A joint work with Z. Bradshaw and A. Farhat.
May 4, 2017
Helena Nussenzveig Lopes :
4 p.m. in SEO 427
Abstract
In this talk we consider the vanishing viscosity problem for incompressible fluid flow in a smooth, bounded domain. We are particularly concerned with the shear layer occurring near the boundary, a phenomenon which we explore in a few examples of flows with symmetry. We use this discussion as motivation for a framework for weak solutions of the inviscid equations which allow for exchange between circulation around the boundary and vorticity in the bulk of the fluid.
Sept. 18, 2017
Pavel Lushnikov :
4 p.m. in SEO 636
Abstract
2D hydrodynamics of ideal fluid with free surface is considered. A
time-dependent conformal transformation is used which maps a free
fluid surface into the real line with fluid domain mapped into the
lower complex half-plane. The fluid dynamics is fully characterized
by the complex singularities in the upper complex half-plane of the
conformal map and the complex velocity. The initially flat surface
with the pole in the complex velocity turns over arbitrary small
time into the branch cut connecting two square root branch points.
Without gravity one of these branch points approaches the fluid
surface with the approximate exponential law corresponding to the
formation of the fluid jet. The addition of gravity results in
wavebreaking in the form of plunging of the jet into the water
surface. The use of the additional conformal transformation to
resolve the dynamics near branch points allows to analyze
wavebreaking in details. The formation of multiple Crapper capillary
solutions is observed during overturning of the wave contributing to
the turbulence of surface wave. Another possible way for the
wavebreaking is the slow increase of Stokes wave amplitude through
nonlinear interactions until the limiting Stokes wave forms with
subsequent wavebreaking. For non-limiting Stokes wave the only
singularity in the physical sheet of Riemann surface is the
square-root branch point located. The corresponding branch cut
defines the second sheet of the Riemann surface if one crosses the
branch cut. The infinite number of pairs of square root
singularities is found corresponding to infinite number of
non-physical sheets of Riemann surface. Each pair belongs to its own
non-physical sheet of Riemann surface. Increase of the steepness of
the Stokes wave means that all these singularities simultaneously
approach the real line from different sheets of Riemann surface and
merge together forming 2/3 power law singularity of the limiting
Stokes wave. It is conjectured that non-limiting Stokes wave at the
leading order consists of the infinite product of nested square root
singularities which form the infinite number of sheets of Riemann
surface. The conjecture is also supported by high precision
simulations, where a quad (32 digits) and a variable precision (up
to 200 digits) were used to reliably recover the structure of square
root branch cuts in multiple sheets of Riemann surface.
Sept. 25, 2017
Radu Dascaliuc :
4 p.m. in SEO 636
Abstract
The idea of re-casting existence and uniqueness for a mild formulation of the Navier-Stokes equations in terms of multiplicative stochastic processes (cascades) goes back to Le Jan and Sznitman work in the 1990’s. In this talk I will address these cascades in the scaling-invariant setting, showing that the process develops infinitely many branches in finite time — a phenomenon called explosion. In previous work, explosion presented an obstacle in establishing uniqueness of the solutions. Nevertheless, we can show that both existence and uniqueness hold for small initial data. I will conclude by discussing possible implications of explosion to the problem of uniqueness of the solutions with large initial data.
Oct. 16, 2017
Hongqiu Chen :
4 p.m. in SEO 636
Abstract
Considered here is a system $U_t+U_x-U_{xxt}+(\nabla H(U))_x=0$ of nonlinear dispersive equations, where $U=U(x,t)$ is an $\mathbb R^2$-valued function, and $\nabla H$ is the gradient of a homogeneous polynomial function $H:\mathbb R^2\rightarrow \mathbb R$ of degree $p\geq 3.$
We present existence of explicit solitary wave solutions. Using the idea by Bona, Chen and Karakashian and exploiting the accurate point spectrum information of the associated Schr$\ddot{o}$dinger operator, we derive a simple algebraic condition for stability of the explicit solitary wave solutions, which
improves the stability results previously obtained by Pereira and also observe the criteria for instability of solitary wave solutions.
Oct. 30, 2017
Javier Gomez-Serrano :
4 p.m. in SEO 636
Abstract
The SQG equation models the formation of fronts of hot and cold air. In a different direction this system was proposed as a 2D model for the 3D incompressible Euler equations. It is not known at this moment if this equation can produce singularities. In this talk I will discuss some recent work on the existence of nontrivial families of global solutions of the SQG equation and related models. Joint work with Angel Castro, Diego Cordoba and Alex Ionescu.
Nov. 6, 2017
Youngjoon Hong :
4 p.m. in SEO 636
Abstract
Numerical methods for the partial differential equations have made enormous progress over the past decades. The needs in mathematical modeling for efficient numerical algorithms as an alternative to classical methods of applied mathematics have remarkably increased. In this talk, recent developments of numerical methods and computations for physical models with a small parameter are presented. More precisely, we explore the numerical methods for convection-dominated singularly perturbed problems and scattering problem in layered media.
Nov. 13, 2017
Lingxing Yao :
4 p.m. in SEO 636
Abstract
Differences in solute concentration across a semipermeable membrane of cells generates
transmembrane osmotic water flow. The interaction of such flows with membrane and flow mechanics is a little explored
area despite its potential significance in many biological applications. Particularly, in recent studies,
experimental evidence suggests that membrane ion channels and aquaporins (water channels),
and thus, solute diffusion and osmosis, play an important role in cell movement.
To clarify the role of osmosis in cell movement, one needs to understand the interplay between solute diffusion, osmosis and mechanical forces.
In this presentation, we discuss a mathematical model that allows for studying the interplay between diffusive, osmotic
and mechanical effects, and the numerical method for solving the model system. An
osmotically active solute obeys a advection-diffusion equation in a region
demarcated by a deformable membrane. The interfacial membrane allows transmembrane water flow
which is determined by osmotic and mechanical pressure differences across the membrane.
The numerical method is based on an immersed boundary method for fluid-structure
interaction and a Cartesian grid embedded boundary method for the solute. We demonstrate our numerical algorithm
with the test case of an osmotic engine, a recently proposed mechanism for cell propulsion.
This is joint work with Yoichiro Mori at University of Minnesota.
Nov. 20, 2017
Misun Min :
4 p.m. in SEO 612
Abstract
Simulations of ion channels and metamaterial devices are of considerable technological importance and also present very interesting and challenging problems from the perspective of numerical algorithms for PDEs. Robust and rapidly convergent numerical methods are essential to solve these systems. I will discuss efficient high-order algorithms for solving drift-diffusion models that describe the transport of charge-carriers coupled with a Poisson equation for electric potential. The spatial discretization is based on a standard spectral-element formulation with body-fitted hexahedral elements. The temporal discretization is a mixed implicit-explicit method using kth-order backward-difference formulas and extralpolation as well as a pseudo-timestepping approach based on Jacobi-free Newton Krylov methods. The electric potential is governed by a Poisson equation to be solved at each timestep. This problem is solved with GMRES iteration, preconditions with spectral-element multigrid. Validation of the algorithms is demonstrated with convergence studies and computational results for potassium ion channels. A second problem of interest is the study of novel ultra-thin flat metalens and graphene-based two-dimensional materials, whose behavior is governed by the solutions to Maxwell's equations. I will discuss high-order spectral element discontinuous Galerkin schemes for solving these equations. High-order discretizations offer minimial numerical dispersion and dissipation while providing high performance to deliver fast, efficient, and accurate simulations on current and next-generation architectures. The discussion will include the aspects of high-performance algorithms and performance analysis on many-core and many-GPU cmputing platforms.
Nov. 27, 2017
Roman Shvydkoy :
4 p.m. in SEO 636
Abstract
We will introduce and study a new class of models describing phenomena of alignment and flocking in systems of collective behavior. The models pose new challenges to the regularity theory of fractional parabolic systems and open a range of new problems in the analysis of PDEs.
Dec. 4, 2017
Chris Henderson :
4 p.m. in SEO 636
Abstract
The Harnack inequality requires one to look back in time to relate the supremum and infimum of a solution to a parabolic equation. In this talk, I will introduce a Harnack-type inequality that allows us to remove this looking-back-in-time restriction at the expense of a slightly weaker bound. I will then discuss applications of this bound to three non-local reaction-diffusion equations arising in biology and combustion. In particular, in each case, this inequality allows us to show that solutions to these equations, which do not enjoy a maximum principle, may be compared with solutions to a related local equation, which does enjoy a maximum principle. Precise estimates of the propagation speed follow from this.
Jan. 29, 2018
Karen Zaya :
2 p.m. in SEO 636
Feb. 19, 2018
Hongjie Dong :
4 p.m. in SEO 636
Abstract
I will present a recent result regarding Dini type estimates for Bellman type fully nonlinear nonlocal parabolic equations with drift terms, and for linear nonlocal parabolic equations with time-irregular data. No smallness condition on the drift coefficients is imposed when the order of the nonlocal operators is greater or equal to 1. Joint work with Tianling Jin (HKUST) and Hong Zhang (Brown University).
Feb. 26, 2018
Xiaoyutao Luo :
4 p.m. in SEO 636
Abstract
In this talk I will present new conditions implying energy equality for weak solutions of 3D Navier-Stokes equations. These conditions are weak-in-time with optimal space regularity and therefore weaker than all previous classical results.
March 5, 2018
Yingwei Wang :
4 p.m. in SEO 636
Abstract
In general, solutions to the Laplacian equation enjoy relatively high smoothness. However, they can exhibit singular behaviors at domain corners or points where boundary conditions change type. In this talk, I will focus on the mixed Dirichlet-Neumann boundary conditions for Laplacian equation, and discuss how singularities in this case adversely affect the accuracy and convergence rates of standard numerical methods. Then, starting from the celebrated Weierstrass theorem about polynomial approximation, I will describe the approximation theory related to the so-called Muntz polynomials, which can be viewed as a generalization of usual polynomials. Additionally, I will illustrate the idea of Muntz-Galerkin methods, and show that how they can overcome the difficulties to achieving high order accuracy for the problems with singularities.
March 12, 2018
Stephen Cameron :
4 p.m. in SEO 636
Abstract
The Muskat problem was originally introduced by Muskat in order to model the interface between water and oil in tar sands. In general, it describes the interface between two incompressible, immiscible fluids of different constant densities in a porous media. In this talk I will prove the existence of global, smooth solutions to the 2D Muskat problem in the stable regime whenever the initial data is monotonic or has slope strictly less than 1. The curvature of these solutions solutions decays to 0 as $t$ goes to infinity, and they are unique when the initial data is $C^{1,\epsilon}$. We do this by constructing a modulus of continuity generated by the equation, just as Kiselev, Naverov, and Volberg did in their proof of the global well-posedness for the quasi-geostraphic equation.
March 19, 2018
Sergey Dyachenko :
4 p.m. in SEO 636
Abstract
We study the problem of 2D incompressible fluid dynamics with free surface, we assume
the the fluid is ideal and the flow is potential. Following the conformal mapping
technique we reformulate the problem to surface variables and demonstrate the existence
of previously undiscovered constants of motion associated with singularities in the
analytic continuation of conformal map and complex potential. In numerical simulations
we recover the analytic structure of the surface shape and observe simple poles and
branch point singularities of the square-root type. We use the Alpert-Greengard-Hagstrom
method to recover the location, type and magnitude of the singularities. We show how
the approach of square-root type singularities may be responsible for the breaking of
waves in the ocean, following the nonlinear stage of modulational instability.
April 2, 2018
Shijun Zheng :
4 p.m. in SEO 636
Abstract
Vortex type solitons exhibit remarkable and ubiquitous phenomena that arise in modeling quantum optics, plasma, superfluids and pseudo-relativistic boson stars.
I will discuss orbital stability and instability by providing certain sharp conditions for the governing equations including magnetic and fractional NLS with unbounded potentials.
This study is motivated by related open question in the area in oder to understand the asymptotic behavior and rates of wave-collapse for the solutions. Some numerical simulations are presented as well.
April 9, 2018
Andreas Kloeckner :
4 p.m. in SEO 636
Abstract
Quadrature by Expansion, or `QBX', is a systematic, high-order approach to singular quadrature that applies to layer potential integrals with general kernels on curves and surfaces. The efficient and accurate evaluation of layer potentials, in turn, is a key building block in the construction of solvers for elliptic PDEs based on integral equation methods. I will present a new fast algorithm incorporating QBX that evaluates layer potentials on and near surfaces in two and three dimensions with user-specified accuracy, along with supporting theoretical and empirical results on complexity and accuracy. A series of examples on unstructured geometry across a variety of applications in two and three dimensions demonstrates the applicability of the method.
April 23, 2018
Rafail Abramov :
4 p.m. in SEO 636
Abstract
I will explain what is happening with the model of gas consisting of hard spheres,
what are the problems with the conventional understanding, how it can be
rectified, and what comes out of it.
April 25, 2018
Maria Schonbek :
3 p.m. in SEO 636
Abstract
TBA
April 30, 2018
Changyou Wang :
4 p.m. in SEO 636
Abstract
In this talk, I will first discuss the initial-boundary value problem of the Ericksen-Leslie system for time-dependent boundary data of the nematic liquid crystal director field.
I will present some result on the global existence in dimension two. Then I will describe an application to the optimal boundary control for such a system in dimension two.
Sept. 24, 2018
Bongsuk Kwon :
4 p.m. in 636 SEO
Abstract
We discuss existence, time-asymptotic behavior, and quasi-neutral limit for the Euler-Poisson equations. Specifically, under the Bohm criterion, we construct the global-in-time solution near the stationary solution of plasma sheath, and also investigate its time-asymptotic behavior and small Debye length limit. If time permits, some key features of the proof and related problems will be discussed. This is joint work with C.-Y. Jung (UNIST) and M. Suzuki (Nagoya Tech.).
Oct. 1, 2018
Jingwei Hu :
4 p.m. in 636 SEO
Abstract
Kinetic equations play an important role in multiscale modeling hierarchy. It serves as a basic building block that connects the microscopic particle models and macroscopic fluid models. Numerically approximating kinetic equations present several difficulties: 1) high-dimensionality (the equation is in phase space); 2) nonlinearity and stiffness of the collision/interaction terms; 3) positivity of the solution (the unknown is a probability density function); 4) consistency to the limiting models; etc. I will start with a brief overview of the kinetic equations including the Boltzmann equation and the Fokker-Planck equation, and then discuss in particular our recent effort of constructing efficient and robust numerical methods for these equations, overcoming some of the aforementioned difficulties. This is joint work with Ruiwen Shu (University of Maryland).
Oct. 8, 2018
Juan Restrepo :
4 p.m. in 636 SEO
Abstract
In the statistics community “Big Data” science is meant to suggest the combining of inferential and computational thinking. We also speak of big data in the geosciences. However, the problems pursued in geoscience are often extreme in the number of degrees of freedom, and in many instances, non-stationary in its statistics. This usually means that we are working with sparse observational data sets, even if the number of observations is large. The Bayesian framework is a natural inferential data assimilation strategy in geosciences, to some extent because the degrees of freedom in the problem vastly outnumber observations but more critically, because the models we use to represent nature have considerable predictive power.Looking toward the future, we expect improvements in computational efficiency and finer resolutions in models, as well as improved field measurements. This will force us to contend with physics and statistics across scales and thus to think of ways to couple multiphysics and computational resolution, as well as to develop efficient methods for adaptive statistics and statistical marginalization.How this coupling is exploited to improve estimates that combine model outcomes and data will be described in tracking hurricanes and improving the prediction of the time and place of coastal flooding due to ocean swells. Estimating the trend of Earth’s temperature from sparse multi-scale data will be used as an example of adaptivity in time series analysis.Other open challenges in non-stationary big data problems will be described, where progress could result from “Big Data Geoscience,” the tighter integration of geoscience, computation, and inference.
Oct. 15, 2018
Qin Li :
4 p.m. in 636 SEO
Abstract
Many kinetic equations have the corresponding fluid limit. In the zero limit of the Knudsen number, one derives the Euler equation out of the Boltzmann equation and the heat equation out of the radiative transfer equation. While there are good numerical solvers for both kinetic and fluid equations, it is not quite well-understood when the two regimes co-exist. In this talk, we model the layer between the fluid and the kinetic using a half-space equation, study the well-posedness, design a numerical solver, and utilize it to couple the two sets of equations that govern separate domains.
Oct. 22, 2018
Fred Weissler :
4 p.m. in 636 SEO
Abstract
We consider the nonlinear heat equation $u_t - \Delta u = |u|^\alpha u$ on ${\mathbb R}^N$, where
$\alpha >0$. It is well known that the Cauchy problem is locally well-posed in a variety of spaces. For
instance, for every $\alpha >0$, it is well-posed in the space $C_0 ( {\mathbb R}^N )$ of continuous
functions that converge to $0$ at infinity. It is also well-posed in $L^p({\mathbb R}^N )$ for $p\ge 1$,
$p>\frac {N\alpha } {2}$, but not well-posed in $L^p$ for $1\le p< \frac {N\alpha } {2}$ if
$\alpha >\frac {2} {N}$. In particular, for such $p$ there exist positive initial values $u_0 \in L^p$ for
which there is no local in time positive solution. Also, if one considers the initial value
$u_0 (x)= c |x|^{-\frac {2} {\alpha }}$ for all $x\in {\mathbb R}^N \setminus \{0\}$, with $c>0$,
it is known that if $c$ is small, there exists a global in time (positive) solution with $u_0$ as initial
value, and in fact this solution is self-similar.
On the other hand, if $c$ is large, there is no local in time positive solution, self-similar or otherwise.
We prove that in the range $0 < \alpha <\frac {4} {N-2}$, for every $c>0$, there exist infinitely many self-similar solutions to the Cauchy problem with initial value $u_0 (x)= c |x|^{-\frac {2} {\alpha }}$. Of course, these solutions are all sign-changing if $c$ is sufficiently large. Also, in the range $\frac {2} {N}< \alpha <\frac {4} {N-2}$, we
prove the existence of local in time sign-changing solutions for a class of nonnegative initial values $u_0 \in L^p$, for $1\le p< \frac {N\alpha } {2}$, for which no local in time positive solution exists.
This is joint work with T. Cazenave, F. Dickstein and I. Naumkin.
Oct. 29, 2018
John Albert :
4 p.m. in 636 SEO
Abstract
We characterize 2-soliton solutions of the Korteweg-de Vries equation as global minimizers for a constrained
variational problem. In particular this gives a nice proof of their stability. The proof uses the so-called profile
decomposition or bubble decomposition of a bounded sequence in Sobolev space.
Nov. 5, 2018
Dana Mendelson :
4 p.m. in 636 SEO
Abstract
In this talk, I will discuss several problems on nonlinear wave and dispersive equations with random initial data, including the energy critical nonlinear wave and Schroedinger equations, and derivative nonlinear wave equations. I will present several almost sure well-posedness and scattering results for these equations and contrast the ways in which random data techniques can be exploited in these different contexts.
Nov. 12, 2018
Yue Yu :
4 p.m. in 636 SEO
Abstract
Bioprosthetic heart valves (BHVs) are the most popular artificial replacements for diseased valves that mimic the structure of native valves. However, the life span of BHVs remains limited to 10-15 years, and the mechanisms that underlie BHVs failure remain poorly understood. Therefore, developing a unifying mathematical framework which captures material damage phenomena in the fluid-structure interaction environment would be extremely valuable for studying BHVs failure. Specifically, in this framework the computational domain is composed of three subregions: the fluid (blood) , the fracture structure (damaged BHVs) modeled by the recently developed nonlocal (peridynamics) theory, and the undamaged thin structure (undamaged BHVs). These three subregions are numerically coupled to each other with proper interface boundary conditions.
In this talk, I will introduce two sub-problems and the corresponding numerical methods we have developed for this multiscale/multiphysics framework. In the first problem the coupling strategy for fluid and thin structure is investigated. This problem presents unique challenge due to the large deformation of BHV leaflets, which causes dramatic changes in the fluid subdomain geometry and difficulties on the traditional conforming coupling methods. To overcome the challenge, the immersogemetric method was developed where the fluid and thin structure are discretized separately and coupled through penalty forces. To ensure the capability of the developed method in modeling BHVs, we have verified and validated this method. In the second problem, we proposed a Neumann-type interface boundary condition for the nonlocal model. In the nonlocal models the Neumann-type boundary conditions should be defined in a nonlocal way, namely, on a region with non-zero volume outside the surface, while in fluid—structure interfaces the hydrodynamic loadings from the fluid side are typically provided on a sharp co-dimension one surface. Therefore, we have shown that our new nonlocal Neumann-type boundary condition provides an approximation of physical boundary conditions on a sharp surface, with an optimal asymptotic convergence rate to the local counter part. Based on this new boundary condition, we have developed a fluid—peridynamics coupling framework without overlapping regions.
Nov. 26, 2018
Ian Tice :
4 p.m. in 636 SEO
Abstract
The contact line problem in interfacial fluid mechanics concerns the triple-junction between a fluid, a solid, and a vapor phase. Although the equilibrium configurations of contact lines have been well-understood since the work of Young, Laplace, and Gauss, the understanding of contact line dynamics remains incomplete and is a source of work in experimentation, modeling, and mathematical analysis. In this talk we consider a 2D model of contact point (the 2D analog of a contact line) dynamics for an incompressible, viscous, Stokes fluid evolving in an open-top vessel in a gravitational field. The model allows for fully dynamic contact angles and points. We show that small perturbations of the equilibrium configuration give rise to global-in-time solutions that decay to equilibrium exponentially fast.
Dec. 3, 2018
Christopher Rycroft :
4 p.m. in 636 SEO
Abstract
An advection–diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow will be presented. By taking advantage of conformal invariance of the model, a numerical method will be introduced that tracks the evolution of the object boundary in terms of a time-dependent Laurent series. Simulations of several dissolving objects will be shown, all of which show collapse to a single point in finite time. The simulations reveal a surprising connection between the position of the collapse point and the initial Laurent coefficients, which was subsequently derived analytically.
Jan. 28, 2019
Yannis Giannoulis :
4 p.m. in 636 SEO
Abstract
In this talk we consider the water wave problem of finite depth as a nonlinear dispersive system. Motivated by this feature, we are interested in the macroscopic dynamics of the envelopes of small, macroscopically amplitude-modulated fixed carrier waves, the latter being plane wave solutions of the linearized problem. More specifically, we want to know whether some sort of interaction of different (modulated) carrier waves can be observed macroscopically. For pure gravity waves such a macroscopic interaction can be observed only for the next-to-leading order corrections of the macroscopic amplitudes and a relevant system of modulation equations is derived. This system is then justified by employing the stability of the original water wave problem, as established by Lannes in his 2013 book. Time permitting, we discuss also the completely different situation in the case of capillary-gravity water waves, where for resonant carrier waves macroscopic interactions can be observed for the leading order amplitudes and where a stability result for the original water wave problem has to take into account the second-order differential operator of the surface tension.
Feb. 4, 2019
Milton Lopes Filho :
4 p.m. in 636 SEO
Abstract
In this talk we explore the Navier, or slip boundary condition for incompressible fluid flows and discuss some of its consequences for
the vanishing viscosity limit.
Feb. 11, 2019
Shawn Walker :
4 p.m. in 636 SEO
Abstract
We consider the generalized Ericksen model of liquid crystals, which is an energy with 8 independent ``elastic'' constants that depends on two order parameters n (director) and s (variable degree of orientation). In addition, we present a new finite element discretization for this energy, that can handle the degenerate elliptic part without regularization, is stable and it Gamma-converges to the continuous energy. Moreover, it does not require the mesh to be weakly acute (which was an important assumption in our previous work). A minimization scheme for computing discrete minimizers will also be discussed.
Furthermore, we include other effects such as weak anchoring (normal and tangential), as well as fully coupled electro-statics with flexo-electric and order-electric effects. We also present several simulations (in 2-D and 3-D) illustrating the effects of the different elastic constants and electric field parameters.
At the end of the talk, we discuss a problem on the packing of DNA inside viral capsids. We show how the generalized Ericksen model can be used to simulate the packing of DNA inside viral capsids, and to estimate packing pressures inside the capsid. This part is joint with Carme Calderer (UMN), Dmitry Golovaty (U. Akron), Javier Arsuaga (U.C. Davis).
Feb. 18, 2019
David Nicholls :
4 p.m. in 636 SEO
Abstract
The faithful modeling of the propagation of linear waves in a layered,
periodic structure is of paramount importance in many branches of the
applied sciences, in particular, in the simulation and design of
multilayer plasmonic devices. In this talk we present a novel
numerical algorithm for the simulation of such problems which is free
of the artificial singularities present in related approaches. We
advocate for a non-overlapping domain decomposition method (DDM)
phrased in terms of Impedance-Impedance Operators that are immune to
the Dirichlet eigenvalues which plague the Dirichlet-Neumann Operators
that appear in classical formulations. We demonstrate a High-Order
Spectral algorithm to simulate these operators based upon a High-Order
Perturbation of Surfaces methodology which is rapid, robust, and
highly accurate. We demonstrate the validity and utility of our
approach with a sequence of numerical simulations.
Feb. 25, 2019
Michael Greenblatt :
4:15 p.m. in 636 SEO
Abstract
We describe some sharp estimates for Fourier transforms of indicator functions of bounded open sets in R^n with real analytic boundary. These estimates are closely connected to corresponding sharp estimates on Fourier transforms of hypersurface measures. The estimates have immediate number theoretical applications, providing nontrivial lattice point discrepancy results for a large class of domains. Unlike most previous results in this subject, no convexity condition is required on the domains.
These estimates also have applications to maximal averages in harmonic analysis and local stability theorems for integrals of negative powers of real-analytic functions, which will be described if time permits.
March 4, 2019
Gideon Simpson :
4 p.m. in 636 SEO
Abstract
Rough energy landscapes appear in a variety of applications including disordered media and soft matter. In this work, we examine challenges to sampling from Boltzmann distributions associated with rough energy landscapes. Here, the roughness will correspond to highly oscillatory, but bounded, perturbations of a fundamentally smooth landscape. Through a combination of numerical experiments and asymptotic analysis, we demonstrate that the performance of Metropolis Adjusted Langevin Algorithm can be severely attenuated as the roughness increases. In contrast, we prove, rigorously, that Random Walk Metropolis is insensitive to such roughness. We also formulate two alternative sampling strategies that incorporate large scale features of the energy landscape, while resisting the impact of roughness; these also outperform Random Walk Metropolis. Numerical experiments on these landscapes are presented that confirm our predictions. Open analysis questions and numerical challenges are also highlighted.
This is joint work with P. Plechac (Delaware).
March 11, 2019
Paula Vasquez :
4 p.m. in 636 SEO
Abstract
Viscoelastic materials are characterized by the coupling of microstructural changes to macroscale deformations. In this talk, we discuss an elastic dumbbell model that leverages the parallel processing power of High Performance Computing (HPC) Graphics Processing Units (GPUs) to create a unique micro-macro scale driven design which incorporates the nonlinear nature of viscoelastic responses as well as the stochastic processes which describe the breaking and reforming of entanglements in the underlying microscopic network. The model allows a full reconstruction of the microstructure-flow coupling thereby creating a platform with the ability to investigate how microscopic changes affect macroscopic responses. Here, we focus on oscillatory flow and show both evolution of stress and species distribution as functions of frequency and strain.
March 18, 2019
Matthew Novack :
4 p.m. in 636 SEO
Abstract
The 3D Quasi-Geostrophic system is a set of equations used in meteorology to describe the evolution of the atmosphere. The surface quasi-geostrophic equation (2D SQG) is a well-studied special case where the atmosphere above the earth is at rest. In this talk, we will discuss a pair of recent results, the first of which derives the physical boundary conditions for the 3D model and constructs global in time weak solutions. The second result shows the non-uniqueness of weak solutions to the 3D model via a convex integration argument.
April 1, 2019
Ryan Obermeyer :
4 p.m. in 636 SEO
Abstract
We study a generalized dissipative Gross-Pitaevskii-type model
arising in the description of exciton-polariton condensates. We derive rigorous
existence and uniqueness results for this model posed on the one dimensional
torus and derive various a-priori bounds on its solution. Then, we analyze
in detail the long time behavior of spatially homogenous solutions and their
respective steady states. In addition, we will present numerical simulations in
the case of more general initial data. We also study the corresponding adiabatic
regime which results in a single damped-driven Gross-Pitaveskii equation and
compare its dynamics to the one of the full coupled system.
Joint work with C. Sparber, P. Antonelli, P. Markowich, and J. Sierra
April 8, 2019
Radu Purice :
4 p.m. in 636 SEO
Abstract
Consider a periodic Schrödinger operator in two dimensions, perturbed by a weak magnetic fi eld whose intensity
slowly varies in space. We show in great generality that the bottom of the spectrum of the corresponding
magnetic Schrödinger operator develops spectral islands separated by gaps, reminicent of a Landau-level
structure.
April 29, 2019
Colette Guillope :
4 p.m. in 636 SEO
May 6, 2019
Theodor Drivas :
4 p.m. in 636 SEO
Abstract
The problem of minimizing energy dissipation and wall drag in turbulent pipe and channel
flows is a classical one which is of great importance in practical engineering applications.
Remarkably, the addition of trace amounts of polymer into a turbulent flow has a pronounced
effect on reducing friction drag. To study this mathematically, we introduce a new boundary
condition for Navier-Stokes equations which models the situation where polymers are irreversibly
grafted to the wall. For engineering applications, the effects of polymer on drag reduction are
thought to be most pronounced near the boundary and therefore such wall-grafted polymer chains
are often employed as drag-reducing agents. Our boundary condition - derived from a fluid-polymer
stress balance - closes in the macroscopic fluid variables and becomes an evolution equation for the
vorticity along the solid walls. We prove global well-posedness for the resulting system in two spatial
dimensions and show that it captures the drag reduction effect in the sense that the vanishing viscosity
limit holds with a rate. Consequently, we obtain bounds on energy dissipation rate and drag which
qualitatively agree with observations of drag reduction in laminar flow.
Talk is based on joint work with Joonhyun La.
Aug. 1, 2019
Eduard Feireisl :
11 a.m. in 636 SEO
Abstract
TBA
Sept. 9, 2019
Shi Jin :
4 p.m. in 636 SEO
Abstract
We develop random batch methods for interacting particle systems with large number of particles. These methods use small but random batches for particle interactions,
thus the computational cost is reduced from O(N^2) per time step to O(N), for a
system with N particles with binary interactions.
For one of the methods, we give a particle number independent error estimate under some special interactions. Then, we apply these methods
to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian motion from random matrix theory, Thomson's problem,
distribution of wealth, opinion dynamics and clustering. Numerical results show that
the methods can capture both the transient solutions and the global equilibrium in
these problems.
Sept. 30, 2019
Daniel Lear Claveras :
4 p.m. in 636 SEO
Abstract
A fluid is said to be in hydrostatic equilibrium when it is at rest. Then the forces acting on it must balance it. A natural question therefore arises:
What happens if our initial data is close to an hydrostatic equilibrium solution?
The field of hydrodynamic stability has a long history starting in the 19th century. For us, the basic problem is to consider a perturbation of the hydrostatic equilibrium, in which case the fluid must start to move, and to study the long-time behavior of the solution. In this talk, we study the stability of the hydrostatic equilibrium in two different problems.
1st: The inviscid incompressible porous media equation.
2nd: The inviscid and non-diffusive Boussinesq system with a velocity damping term.
Oct. 7, 2019
Alexander Watson :
4 p.m. in 636 SEO
Abstract
In the simplest case, topological insulators are two-dimensional materials which support robust, one-way, edge currents. Despite recent progress on rigorously understanding edge currents through the ``bulk-boundary correspondence’’ principle, basic practical questions about 2d materials, which are important for understanding topological insulators, remain to be answered. In this talk I will present (1) a new numerical method for computing edge states of 2d materials in the presence of edge defects, and (2) a new numerical method for computing Wannier functions of 2d materials which suggests a generalization of the ``localization-topology’’ dichotomy to general disordered (without periodic structure) 2d materials.
Oct. 14, 2019
Matthias Maier :
4 p.m. in 636 SEO
Abstract
In the terahertz frequency range, the effective (complex-valued) surface
conductivity of atomically thick 2D materials such as graphene has a
positive imaginary part that is considerably larger than the real part.
This feature allows for the propagation of slowly decaying electromagnetic
waves, called surface plasmon-polaritons (SPPs), that are confined near the
material interface with wavelengths much shorter than the wavelength of the
free-space radiation. SPPs are a promising ingredient in the design of novel
optical devices, promising "subwavelength optics" beyond the diffraction
limit. There is a compelling need for controllable numerical schemes which,
placed on firm mathematical grounds, can reliably describe SPPs in a
variety of geometries.
In this talk we present a number of analytical and computational approaches
to simulate SPPs on 2D material interfaces and layered heterostructures.
Aspects of the numerical treatment such as absorbing perfectly matched
layers, local refinement and a-posteriori error control are discussed. We
show analytical results for some prototypical geometries and a
homogenization theory for layered heterostructures.
Oct. 21, 2019
Fred Weissler :
4 p.m. in 636 SEO
Abstract
Abstract : We consider a nonlinear heat equation on $R^n$ with a homogeneous, superlinear nonlinearity.
We study solutions which are anti-symmetric with respect to the spatial variables.
It is shown that very singular initial values, e.g. derivatives of the Dirac delta function,
give rise to local (regular) solutions. These solutions exist when the homogeneity of the nonlinearity
is below a value which is consistent with the scaling properties of the equation.
These results enable us to obtain new finite time blowup results for certain classes
of regular anti-symmetric initial values of the form $u_0 = \lambda f$.
Counterintuitively, these blow-up results hold for $\lambda > 0$ sufficiently small.
Blowup results of this type, where the initial value has a small coefficient, were
first found by Dickstein [1]. The work to be presented is joint with S. Tayachi [2, 3].
References
[1] F. Dickstein, Blowup stability of solutions of the nonlinear heat equation with a large life span,
J. Di erential Equations 223 (2006), 303{328.
[2] S. Tayachi and F. B. Weissler, The nonlinear heat equation with high order mixed derivatives
of the Dirac Delta as initial values, Trans. Amer. Math. Soc. 366 (2014), 505-530.
[3] S. Tayachi and F. B. Weissler, The nonlinear heat equation involving highly singular initial
values and new blowup and life span results, Journal of Elliptic and Parabolic Equations, 4
(2018), 141-17
Nov. 18, 2019
Ying Wang :
4:15 p.m. in 636 SEO
Abstract
In this talk, I will discuss a new class of entropy solutions of the modified Buckley-Leverett equations, which model underground oil recovery. Analytic study on the computational domain reduction and traveling wave solution stability will be provided. A variety of numerical examples will be given. They show that the solutions may have many different profiles depending on the initial conditions, diffusion parameter, and the third-order mixed derivatives parameter. The results are consistent with the study of traveling wave solutions and their bifurcation diagrams.
Nov. 20, 2019
Christian Klein :
3 p.m. in 636 SEO
Abstract
We present a survey of numerical studies on solutions to equations from the family of Korteweg-de Vries (KdV) equations. We concentrate on the appearance of solitons in the long time behavior of the solutions, zones of rapid modulated oscillations called dispersive shock waves, and of a loss of regularity (blow-up) in finite time. In particular we study in 1d generalized KdV and fractional KdV equations. In 2d we study Kadomtsev-Petviashvili and Zakharov-Kuznetsov equations.
Nov. 25, 2019
Anne Bronzi :
4 p.m. in 636 SEO
Abstract
In this talk we will explore the regularity of viscosity solutions for a class of variable-exponent, degenerate/singular elliptic equations in non-divergence form. More precisely, we will prove that viscosity solutions to the equation $|Du|^{p(x)}F(D^2u) = f(x)$, where $F$ is a uniformly elliptic operator and $p$ satisfies mild conditions, are locally of class $C^{1,\alpha}$.
This is joint work with E. Pimentel (PUC-Rio), G. Rampasso (Unicamp) and E. Teixeira (UCF).
Nov. 26, 2019
Angel Castro :
3 p.m. in 636 SEO
Abstract
We will present a family of equations which give rise to particular solutions of the Euler-Alignment system. This system is the macroscopic version of the Cucker-Smale model which is concerned with the motion of a collection of agents. We will show that, under suitable assumptions on the initial data the solutions form singularities in finite time. Joint work with Victor Arnaiz.
Dec. 2, 2019
Paul Plucinsky :
4:15 p.m. in 636 SEO
Abstract
Pattern formation at microscopic scales is an ubiquitous property of active materials undergoing
phase transformation or mechanical loading. In shape memory alloys, for instance, a temperature
change beyond a critical temperature can result in the branching of fine-scale twins of the low temperature phase (martensite) at an interface with the high temperature phase (austenite).
Additionally, in nematic elastomer sheets, loading transverse to the sheet's natural orientational order induces fine-scale oscillations in this order and a soft elastic response. In this talk, I will discuss
how the aforementioned features can be modeled by variational methods, and how this modeling
can be used to make predictions of the material behavior at the engineering scale.
Feb. 10, 2020
Nadia Drenska :
4 p.m. in 636 SEO
Abstract
Prediction with expert advice is an area of online machine learning, which aims to synthesize advice from different experts. We consider the case of a stock prediction problem with an investor who relies on history-dependent experts, and an adversarial market. This forms a two-person game, and we are interested in the optimal strategies of the market and the player when the game is played over a long time. We prove that the discrete value function converges to the unique solution of a nonlinear parabolic PDE, which determines asymptotically optimal strategies.
Feb. 24, 2020
David Goluskin :
4 p.m. in 636 SEO
Abstract
Many complex systems are governed by nonlinear ODEs or PDEs that cannot be solved exactly. Various properties of such solutions can be inferred by constructing auxiliary functions satisfying suitable inequalities. The most familiar example is the construction of Lyapunov functions to infer stability of particular states, but similar approaches can produce many other types of mathematical statements, including for systems with chaotic or otherwise complicated behavior. Such statements include estimates of time-averaged quantities and extreme transient behavior, approximation of nonlinear stability properties, and design of controls. In many cases, the search for the auxiliary function that implies the strongest mathematical statement can be posed as a convex optimization problem. Such problems can be studied analytically or computationally, but in most cases computation is needed to find solutions that are close to optimal. Of particular use are computational methods of polynomial optimization, where the optimization constraints include polynomial inequalities. This talk will provide an overview of different ways in which auxiliary functions can be used to study nonlinear ODEs and PDEs, as well as how polynomial optimization can be used to implement these methods computationally. Methods will be illustrated using applications to various complex systems.
March 2, 2020
Benoit Pausader :
4 p.m. in 636 SEO
Abstract
The Euler-Maxwell equations model the behavior of a plasma as the superposition of two compressible charged fluids interacting with their self-consistent electromagnetic field. In the absence of the coupling, small disturbances can create shocks in compressible Euler equations, but we show that this instability disappears with the electromagnetic field. This reduces to a small data/global existence result for quasilinear dispersive systems. I will review different joint works with Y. Deng, Y. Guo, A. Ionescu, E. Grenier and M. Suzuki.
March 16, 2020
David Ambrose :
4 p.m. in 636 SEO
Abstract
When considering N-player differential games, making the approximation that there are instead infinitely many agents leads to the mean field games system of PDEs. This system has two unknowns, the probability distribution of the players, and the value function being optimized by a representative agent. One of these satisfies a forward parabolic equation and the other satisfies a backward parabolic equation. The forward parabolic equations comes with initial data while terminal data (at a fixed time T>0) is specified for the backward parabolic equation. We will describe some existence results for this coupled forward-backward system, including for a specific system which has been given as a model of household wealth.
March 30, 2020
Sergey Nadtochiy :
4 p.m. in 636 SEO
April 6, 2020
Ohannes Karakashian :
4 p.m. in 636 SEO
Abstract
We construct a priori and a posteriori error estimates for a system of KdV-KdV equations coupled
through their nonlinear terms. We present numerical experiments validating the theoretical results.
In particular, stability of solitary-wave solutions is investigated and reveals remarkable differences
when compared with similar theory for a (single) KdV equation.
April 8, 2020
Anudeep Kumar Arora :
2 p.m. in 636 SEO
April 13, 2020
Luka Pocivavsek :
4 p.m. in 636 SEO
April 20, 2020
Carme Calderer :
4 p.m. in 636 SEO
April 27, 2020
Yan Guo :
4 p.m. in 636 SEO
Oct. 19, 2020
Susan Friedlander :
4 p.m. in Zoom
Abstract
Many equations that model fluid behavior are derived from systems that encompass multiple physical forces. When the equations are written in non dimensional form
appropriate to the physics of the situation, the resulting PDEs often involve multiple non-dimensional parameters. Frequently some of these parameters are very small and they
enter into the analysis in different ways. We will discuss one such system which has been proposed as a model for magnetostrophic turbulence and describe results that can be
obtained in several different small parameter limits. In this talk we will concentrate on a forced drift-diffusion equation for the temperature where the fluid viscosity enters via the
drift velocity. We examine the convergence of solutions in the limit as the viscosity goes to zero. We introduce a natural notion of ”vanishing viscosity” weak solutions and prove the existence of a compact global attractor for the critical drift-diffusion equation.
This is joint work with Anthony Suen.
Nov. 16, 2020
Anudeep Kumar Arora :
4 p.m. in Zoom
Abstract
In 1924, Louis De Broglie proposed in his PhD thesis the theory of matter waves. On January 26, 1926, Erwin Schrödinger published his work in which he constructed the wave equation for de Broglie's matter waves, now known as the Schrödinger equation. Schrödinger equations are partial differential equations of dispersive category.
In this work, we study the generalized Hartree (gHartree) equation, which is a nonlinear Schrödinger type equation with a nonlocal nonlinearity, of a convolution type. We first, in the energy-subcritical regime, classify the behavior of finite energy solutions under the mass-energy assumption, identifying the sharp threshold for global (scattering) versus finite time (blow-up) solutions. We exhibit two methods of obtaining scattering: one via Kenig-Merle concentration - compactness and another one is using Dodson-Murphy approach via Morawetz estimate and Tao's scattering criteria. Next, we are interested in the phenomenon of wave collapse (blow-up), for which, we investigate stable singularity formations in the mass-critical gHartree equation and rigorously prove a stable blow-up formation in dimension 3.
Nov. 30, 2020
Daniel Lear Claveras :
4 p.m. in Zoom
Abstract
In this work, we study a new class of alignment models with self-propulsion and Rayleigh-type friction forces, which describes the collective behavior of agents with individual characteristic parameters. We describe the long-time dynamics via a new method which allows to reduce analysis from the multidimensional system to a simpler family of two-dimensional systems parametrized by a proper Grassmannian. With this method we demonstrate exponential alignment for
a large (and sharp) class of initial velocity configurations confined to a sector of opening less than $\pi$. In the case when characteristic parameters remain frozen, the system governs dynamics of opinions for a set of players with constant convictions. Viewed as a dynamical non-cooperative game, the system is shown to possess a unique stable Nash equilibrium, which represents a settlement of opinions most agreeable to all agents. Such an agreement is furthermore shown to be a global attractor for any set of initial opinions. Joint work with David N. Reynolds & Roman Shvydkoy
Jan. 11, 2021
Wojciech Ożański :
4 p.m. in Zoom
Abstract
We will consider weak solutions to the 3D incompressiblehypodissipative Navier--Stokes equations (HNSE), $\partial_t u + (- \Delta )^s u + (u\cdot \nabla )u +\nabla p =0$, $\mathrm{div}\, u=0$, where $(-\Delta )^s$ is the Fourier multiplier with symbol $|\xi |^{2s}$, and $s\in (3/4,1)$. We will discuss a new iteration scheme that allows one to study suitable weak solutions of HNSElocally in space-time. We will show how this can be used to obtain that $\nabla^k u \in L^{p,\infty }_{\mathrm{loc}}(\mathbb{R}^3\times(0,\infty))$ for every such solution $u$, where $p=\frac{2(3s-1)}{k+2s-1}$, $k=1,2$, and to improve the partial regularity result of Tang \& Yu (2015) as well provide an estimate on the box-counting dimension of the singular set $S$, $d_B(S\cap \{t\geq t_0 \} )\leq \frac13 (15-2s-8s^2) $ for every $t_0>0$. This is joint work with Hyunju Kwon(IAS).
Feb. 1, 2021
Annalaura Stingo :
4 p.m. in Zoom
Abstract
In this talk we discuss the almost-global well-posedness of a wide class of coupled Wave-Klein-Gordon equations in 2+1 space-time dimensions, when initial data are assumed to be small and localized. The Wave-Klein-Gordon systems arise from several physical models especially related to General Relativity, but few results are know at present in lower space-time dimensions. Compared with prior related results, our novel contributions include a strong quadratic quasilinear coupling between the wave and the Klein-Gordon equation, and no restriction is made on the support of the initial data which are supposed to only have a mild decay at infinity and very limited regularity. Our proof relies on a combination of energy estimates localized to dyadic space-time regions, and pointwise interpolation type estimates within the same regions. This is akin to ideas previously used by Metcalfe-Tataru-Tohaneanu in a linear setting, and is also related to Alinhac’s ghost weight method. A refinement of these estimates through different techniques will allow us to pass, in a future work, from almost global existence to global existence of solutions under the same hypothesis on the initial data. This is a joint work with M. Ifrim.
March 8, 2021
Claire Lestringant :
4 p.m. in Zoom
Abstract
Slender structures are subject to various localised instabilities: necking of bars under
traction, bulging of cylindrical party balloons, beading in cylinders made up of soft gels, or
folding of tape-springs. In all these examples, distinct states of deformation may coexist and
classical one-dimensional (1D) models predict singular solutions. In particular, classical 1D
models fail to describe interfaces or finite size effects. The most common remedy is to use
full structural models based on 3D finite elasticity or nonlinear shell/membrane equations.
However, this is computationally costly and often impracticable: simpler 1D regularised
models depending on the strain and the strain gradient are therefore attractive.
There is a recent effort to rigorously establish 1D higher-order models for the analysis
of localisation in slender structures. I will introduce a systematic method to derive such
models by a formal expansion, starting from a variety of full structural models for slender
elastic structures. The expansion is performed near a finitely pre-strained state and therefore
retains all sources of nonlinearity, coming from the geometry and the constitutive law. I
will illustrate the method in the case of bulging and beading and demonstrate its accuracy
by comparing solutions of the 1D gradient model with solutions of the original structural model.
March 15, 2021
Vincent Martinez :
4 p.m. in Zoom
Abstract
We discuss the existence, uniqueness, and regularity of invariant measures for the damped-driven stochastic Korteweg-de Vries equation, where the noise is additive and sufficiently non-degenerate. It is shown that a simple, but versatile control strategy, typically employed to establish exponential mixing for strongly dissipative systems such as the 2D Navier-Stokes equations, can nevertheless be applied in this weakly dissipative setting to establish both unique ergodicity, albeit without mixing rates, as well as regularity of the support of the invariant measure. Under the assumption of large damping, however, exponential mixing can be recovered.
March 29, 2021
Youngjoon Hong :
4 p.m. in Zoom
Abstract
Deep neural networks have achieved state-of-the-art performance in a variety of fields. The exponential growth of machine learning models and the extreme success of deep learning have seen application across a multitude of disciplines. Recent works observe that a class of widely used neural networks can be viewed as the Euler method of numerical discretization. From the numerical discretization perspective, Total Variation Diminishing (TVD) Runge-Kutta methods are more advanced techniques than the explicit Euler method that produce both accurate and stable solutions. Motivated by the TVD property and a generalized Runge-Kutta method, we proposed new networks which improve robustness against adversarial attacks. If time permits, we explore a deep learning methodology that can be applied to the data-driven discovery of numerical PDEs.
April 5, 2021
David Padilla-Garza :
4 p.m. in Zoom
Abstract
We work with Coulomb gases at an intermediate temperature regime. We define a local empirical field and identify a critical temperature scaling. We show that if the scaling of the temperature is supercritical, the local empirical field satisfies an LDP with an entropy-based rate function. We also show that if the scaling of the temperature is subcritical, the local empirical field satisfies an LDP with an energy-based rate function. An important idea in this work is to exploit the different scaling relations satisfied by the Coulomb energy and the entropy.
April 12, 2021
Maria Pia Gualdani :
4 p.m. in Zoom
Abstract
Kinetic equations are used to describe evolution of interacting particles. The most famous kinetic equation is the Boltzmann equation: formulated by Ludwig Boltzmann in 1872, this equation describes motion of a large class of gases. Later, in 1936, Lev Landau derived a new mathematical model for motion of plasma. This latter equation was named the Landau equation. One of the main features of the Landau equation is nonlocality, meaning that particles interact at large, non-infinitesimal length scales. Moreover, the coefficients are singular and degenerate for large velocities. Many important questions, such as whether or not solutions become unbounded after a finite time, are still unanswered due to their mathematical complexity.
In this talk we concentrate on the mathematical results of the homogeneous Landau equation. We will first review existing results and open problems and in the second part of the talk we will focus on recent developments of well-posedness and regularity theory. This is a joint work with Nestor Guillen.
April 19, 2021
Changhui Tan :
4 p.m. in Zoom
Abstract
How roots of a polynomial evolve under repeated differentiation is a classical question, which goes back to Gauss, Lucas, Marcel Riesz and many others.
The dynamics can be described by a partial differential equation, formally derived by Stefan Steinerberger recently.
The process is also closely related to free probability and random matrices, attracting lots of attention in the last few years.
In this talk, I will explain the intriguing relationship between the roots of trigonometric polynomials under differentiation and Steinerberger's PDE.
The PDE is an active scalar equation, which is closely related to many classical models in fluid dynamics.
In particular, the flow of polynomial roots bears a strong resemblance to self-organized dynamics in the modeling of animal swarms.
The celebrated flocking behavior is connected to the "crystallization" phenomenon on the polynomial root distribution.
I will discuss joint work with Alexander Kiselev on global wellposedness and asymptotic behavior of the PDE, as well as its rigorous connections towards the flow of polynomial roots.
April 26, 2021
Nathan Glatt-Holtz :
4 p.m. in Zoom
Abstract
We treat a statistical regularization of the ill-posed inverse problem of estimating a divergence free flow field $u$
from the partial and noisy observation of a passive scalar $\theta$ which is advected by $u$. Our solution is a Bayesian
posterior distribution, that is a probability measure $\mu$ of the space of divergence free flow fields which
precisely quantifies uncertainties in $u$ once one specifies models for measurement error and a prior knowledge
for $u$. In this talk we survey some of our recent work which analyzes $\mu$ both analytically and numerically.
In particular we discuss a posterior contraction (consistency) result as well as some Markov Chain Monte Carlo
(MCMC) algorithms which we have developed, refined and rigorously analyzed to effectively sample from $\mu$.
This is joint work with Jeff Borggaard, Justin Krometis and Cecilia Mondaini.
Sept. 13, 2021
Rafail Abramov :
4 p.m. in Zoom
Abstract
Conventional models of fluids (e.g. Euler and Navier-Stokes equations) are derived from the Boltzmann equation, where the particle
interactions are described by the collision integral. It is known that particles in a fluid interact deterministically via a potential, yet the
collision integral describes a stochastic, time-irreversible process. Here I show what happens to the fluid mechanics equations when the potential
interaction is not replaced by the collision integral, and some interesting consequences thereof.
Sept. 20, 2021
Jean-Luc Thiffeault :
4 p.m. in 636 SEO
Abstract
A Brownian rod is a long, thin particle undergoing translational and rotational diffusion, due for instance to thermal fluctuations in an ambient fluid. We consider such a rod in a regular lattice of point obstacles. The rod interacts with the obstacles, in the sense that it reflects off of them. An interesting question is then to compute the effective diffusion constant for the rod. I will show how this problem can be partially mapped to a heat conduction problem in a porous medium solved by Rayleigh, and discuss other limits of interest, such as that of anisotropic diffusivities. I will also briefly discuss ongoing work for a swimming organism. This is joint work with Hongfei Chen and Ziheng Zhang.
Oct. 4, 2021
Jason Murphy :
4 p.m. in Zoom
Abstract
We will discuss some recent results on scattering for the cubic-quintic nonlinear Schrödinger equation in two and three space dimensions. We will focus on the problem of scattering at or near certain thresholds, including the sharp mass threshold in the 2d setting and the virial threshold in the 3d setting.
Oct. 11, 2021
Xuenan Li :
4 p.m. in 636 SEO
Abstract
Mechanism-based metamaterials are synthetic materials that exhibit microscale buckling in response to mechanical deformation. Our research focuses on a specific example: the kagome metamaterial. This periodically arranged material has many energy-free buckling patterns without loads on boundaries. In this talk, we will discuss the large-scale behavior of the kagome metamaterial as a nonlinear homogenization problem. The macroscopic theory reveals that only compressive conformal maps achieve zero effective energy. We will also discuss the adequacy of our macroscopic theory with various numerical experiments. The theory is joint work with Robert Kohn, and the numerical results are joint work with Katia Bertoldi and Bolei Deng.
Oct. 18, 2021
Ruiwen Shu :
4 p.m. in Zoom
Abstract
Erdős and Turán proved a classical inequality on the distribution of roots for a complex polynomial in 1950, depicting the fundamental interplay between the size of the coefficients of a polynomial and the distribution of its roots on the complex plane. Various results have been dedicated to improving the constant in this inequality, while the optimal constant remains open. In this paper, we give the optimal constant, i.e., prove the sharp Erdős-Turán inequality. To achieve this goal, we reformulate the inequality into an optimization problem, whose equilibriums coincide with a class of energy minimizers with the logarithmic interaction and external potentials. This allows us to study their properties by taking advantage of the recent development of energy minimization and potential theory, and to give explicit constructions via complex analysis. Finally the sharp Erdős-Turán inequality is obtained based on a thorough understanding of these equilibrium distributions.
Oct. 25, 2021
Caroline Moosmueller :
4 p.m. in 636 SEO
Abstract
Detecting differences and building classifiers between distributions, given only finite samples, are important tasks in a number of scientific fields. Optimal transport (OT) has evolved as the most natural concept to measure the distance between distributions, and has gained significant importance in machine learning in recent years.
There are some drawbacks to OT: Computing OT is usually slow, and it often fails to exploit reduced complexity in case the family of distributions is generated by simple group actions. In this talk, we discuss how optimal transport embeddings can be used to deal with these issues, both on a theoretical and a computational level.
In particular, we'll show how to embed the space of distributions into an L^2-space via OT, and how linear techniques can be used to classify families of distributions generated by simple group actions in any dimension. The proposed framework significantly reduces both the computational effort and the required training data in supervised settings. We demonstrate the benefits in pattern recognition tasks in imaging and provide some medical applications.
Nov. 1, 2021
Siming He :
4 p.m. in Zoom
Abstract
In this talk, we consider the enhanced dissipation phenomena induced by shear flows. In the first part of the talk, I will introduce the idea of shear flow-induced enhanced dissipation and the recent developments on this topic. Then I will exhibit the applications of this phenomenon in various settings, ranging from suppression of chemotactic blow-ups to enhancement of chemical reactions.
Nov. 8, 2021
Chris Rycroft :
4 p.m. in 636 SEO
Abstract
Conventional computational methods often create a dilemma for fluid–structure interaction problems. Typically, solids are simulated using a Lagrangian approach with grid that moves with the material, whereas fluids are simulated using an Eulerian approach with a fixed spatial grid, requiring some type of interfacial coupling between the two different perspectives. Here, a fully Eulerian method for simulating structures immersed in a fluid will be presented. By introducing a reference map variable to model finite-deformation constitutive relations in the structures on the same grid as the fluid, the interfacial coupling problem is highly simplified. The method is particularly well suited for simulating soft, highly-deformable materials and many-body contact problems, and several examples in two and three dimensions will be presented.
Nov. 15, 2021
Oscar Riaño :
4 p.m. in Zoom
Abstract
The study of the competition between dispersive and nonlinearity effects has developed a wide range of questions and applications. In general, the dispersion is fixed, and a variable nonlinearity is considered. An inherent problem in this approach is the study of nonlinearities that involve low regularity, where usually the classical methods of existence cannot be applied directly. In this talk, we will discuss some applications of the approach proposed by Cazenave and Naumkin for the Schrödinger equation, in which the existence of solutions for nonlinearities with low regularity is proved by considering a class of initial data with a certain polynomial behavior. We will focus on the generalized Hartree equation with nonlinearities $p<2$, and a generalized version of the Korteweg–de Vries (KdV) equation.
Nov. 22, 2021
Hongqiu Chen :
4 p.m. in 636 SEO
Abstract
The BBM equation posed on R, R+ and bounded interval, is revisited. Improving on earlier results, global well-posedness and bounds for the growth in time of relevant norms of solutions corresponding to very general auxiliary data are derived. It is based on some simple variable change. Similarly, one can obtain better results for variety of BBM-KdV-type equations.
Nov. 29, 2021
Antonio De Rosa :
4 p.m. in Zoom
Abstract
I will present a $C^{1,\alpha}$-regularity theorem for m-dimensional Lipschitz graphs with anisotropic mean curvature bounded in $L^p$, $p > m$, in every dimension and codimension.
Jan. 24, 2022
Silas Alben :
4 p.m. in Zoom
Abstract
We discuss two problems dealing with the motions of thin deformable bodies under frictional forces. In the first problem, elastic filaments are confined within slowly-shrinking circular boundaries. The filaments undergo deformations that are a mixture of spiraling and bifurcations, primarily the former with small friction and the latter with large friction. With zero friction, a simple model predicts that the maximum curvature and the total elastic energy scale as the wall radius to the -3/2 and -2 powers respectively. With nonzero friction, the elastic energy follows a similar scaling but with a prefactor up to 8 times larger, due to delayering and bending with a range of small curvatures. The second problem examines models of snakes as thin filaments that deform and locomote due to friction. We examine optimal motions of two-link, three-link, and smooth bodies with a variety of friction coefficients. With large friction transverse to the snake, the optimal motion is a retrograde traveling wave with amplitude scaling as the friction coefficient the -1/4 power. With zero transverse friction, a triangular direct wave is optimal. Between these extremes we find a variety of local optima including standing waves (or ratcheting motions).
Jan. 31, 2022
Evelyn Richman :
4 p.m. in 636 SEO
Abstract
We study the strong magnetic field limit for a nonlinear Iwatsuka-type model, i.e. a nonlinear Schr\"odinger equation in two
spatial dimensions with a magnetic vector potential that only depends on the $x$-coordinate. Using a high-frequency averaging technique, we show that this equation can be effectively described by a nonlocal nonlinear model, which is no longer dispersive.
We also prove that, in this asymptotic regime, inhomogeneous nonlinearities are confined along the $y$-axis.
Feb. 7, 2022
Paolo Piersanti :
4 p.m. in Zoom
Abstract
In this talk we discuss the convergence of a numerical scheme, based on the Finite Element Method, via which the solution of an obstacle problem for linearly elastic shallow shells is approximated. For doing so, one needs to establish an augmentation of regularity result for the solution of the problem under consideration. Since the solution is expressed in the form of a vector field, the method proposed by Brezis & Stampacchia appears not to be applicable, and a different approach is thus demanded.
This is joint work with Xiaoqin Shen (Xi’an University of Technology, China).
Feb. 14, 2022
Anuj Kumar :
4 p.m. in 636 SEO
Abstract
We consider the variational problem of "optimal wall-to-wall transport", in which we look to maximize the transport of a passive temperature field between hot and cold plates. Specifically, we optimize the choice of divergence-free velocity field in the advection-diffusion equation, amongst all velocities satisfying an enstrophy constraint. Previous work established an a priori upper bound to transport, scaling as the 1/3 power of the flow’s enstrophy (i.e., the power used to make the flow). Recently, Tobasco and Doering '17 constructed self-similar two-dimensional branching flows saturating this bound up to an unknown logarithmic correction to scaling. We present a three-dimensional "branching pipe flow" that eliminates the possibility of this logarithmic correction, and therefore identifies the optimal scaling as a clean 1/3 power law. Our flows resemble previous numerical studies of the three-dimensional wall-to-wall problem (Motoki, Kawahara and Shimizu ’18), but actually we show using a time-dependent version of our construction that the 1/3 scaling is in fact optimal in two-dimensions as well. These results have natural connections to the outstanding problem of Rayleigh--Bénard convection, and we propose several conjectures along these lines.
Feb. 21, 2022
Yu Gu :
4 p.m. in Zoom
Abstract
I will present a recent work with Tomasz Komorowski and Alex Dunlap in which we derived optimal variance bounds on the solution to the KPZ equation on a large torus, in certain regimes where the size of the torus increases with time. We only use stochastic calculus and I will try to give a heuristic explanation of the 2/3 and 1/3 exponents in the 1+1 KPZ universality class.
Feb. 28, 2022
Brian Seguin :
4 p.m. in 636 SEO
Abstract
In this talk I will mainly discuss two projects related to developable surfaces. The first deals with deforming an unstretchable material surface to form a closed ribbon. The bending energy associated with such a deformation is proportional to the integral of the square of the mean curvature over the deformed surface. However, since the material is unstretchable, this energy can be dimensionally reduced. While this has been recognized in earlier works, I’ll fill in the gaps and present the complete variational problem. The next project, which was inspired by the first, answers the question of how can you construct a developable surface from a space curve. It turns out that besides the well-known tangent and rectifying developable surfaces, there is an entire family of such surfaces that can be generated. The last few minutes of my talk will be dedicated to my current work on extending the dimensional reduction argument of the first part of my talk to more general domains.
March 7, 2022
Daesung Kim :
4 p.m. in 636 SEO
Abstract
Gundy and Varopoulos introduced the probabilistic representation of singular integrals and Fourier mulitpliers such as Hilbert transforms and Riesz transforms as conditional expectations of some stochastic integrals. Combining with the sharp martingale inequalities by Burkholder and Banuelos-Wang, the representations have played a crucial role in finding the sharp, or nearly sharp, $L^p$-bounds for these operators in a variety of geometric settings. Motivated by a recent breakthrough by Banuelos and Kwasnicki on the sharp $\ell^p$-norm of the discrete Hilbert transform, we construct a natural collection of discrete operators in $\mathbb{Z}^{d}$ which have $\ell^p$-norms independent of the dimension. This collection of discrete operators include the probabilistic discrete Riesz transforms, which is the analogues of the probabilistic discrete Hilbert transform used in the paper by Banuelos-Kwasnicki. We also discuss related open problems for the $\ell^p$-norm of discrete operators. This is based on joint work with Rodrigo Banuelos and Mateusz Kwasnicki.
March 14, 2022
Hao Shen :
4 p.m. in 636 SEO
Abstract
We will first review the lattice Yang-Mills model, also called the lattice gauge theories, originally introduced by Wilson as a discretization of quantum Yang-Mills. The model is specified by a well-defined probability measure on a Lie group, at least on finite lattice. We then derive a system of Lie group valued stochastic differential equations, such that the measure is invariant (i.e. Langevin dynamic). We then prove several properties about the model using or related with the dynamic, such as master loop equations, large N factorization, log-Sobolev inequality, and ergodicity. Based on joint work with Scott Smith, Rongchan Zhu, Xiangchan Zhu.
March 28, 2022
Benjamin Dodson :
4 p.m. in Zoom
Abstract
In this talk we will discuss global well-posedness and scattering results for the defocusing, nonlinear Schrödinger equation with initial data in a Besov space. The Besov space is critical under the scaling symmetry. Additionally, we obtain scattering size bounds that are polynomially dependent upon the critical Besov norm.
April 4, 2022
Trevor Leslie :
4 p.m. in 636 SEO
Abstract
The Euler Alignment system is a hydrodynamic version of the celebrated Cucker--Smale ODE's of collective behavior. It can have a hyperbolic or parabolic character, depending on the specified nonlocal interaction protocol; this talk concerns the hyperbolic case in 1D. It is well-established that solutions may lose regularity in finite time, but it has been unknown until recently how to continue to evolve the dynamics after a blowup. After brief orientation on the special structure of these equations, I will describe a recent joint work with Changhui Tan (University of South Carolina), where we developed a theory of weak solutions. Inspired by Brenier and Grenier's work on the pressureless Euler equations, we show that the dynamics of our system are captured by a nonlocal scalar balance law. We generate the unique entropy solution of a discretization of this balance law by introducing the 'sticky particle Cucker--Smale' system to track the shock locations. Our approximation scheme for the density converges in the Wasserstein metric; it does so with a quantifiable rate as long as the initial velocity is at least Hölder continuous.
April 11, 2022
Guher Camliyurt :
4 p.m. in 636 SEO
Abstract
In this talk we will consider the wave equation in odd space dimensions with energy-supercritical nonlinearity, in the radial setting. We will review the concentration compactness and rigidity arguments starting from the earlier work by Kenig-Merle and Duyckaerts-Kenig-Merle in the energy-critical and energy-supercritical cases, and outline the key ideas behind the proof of global well-posedness and scattering results in dimensions three, five, and seven.
April 18, 2022
Oussama Landoulsi :
4 p.m. in Zoom
Abstract
In this talk, we will study the influence of the underlying space geometry on the asymptotic dynamics of the nonlinear Schrödinger (NLS) equation. We will consider the focusing NLS equation in the exterior of a smooth, compact, and convex obstacle with Dirichlet boundary conditions. We will study the asymptotic behavior of the solution for large times and finite time. We prove the existence of these 3 types of solutions: Solitary wave solutions (solitons), blow-up solutions (solutions with finite time of existence) and scattering solutions (global and behaving asymptotically as linear solutions), for the NLS equation in the exterior of a convex obstacle.
April 25, 2022
Francisco Mengual :
4 p.m. in 636 SEO
Abstract
In this talk we consider two problems related to turbulence: The vortex sheet problem for the incompressible Euler equation (Kelvin-Helmholtz instability) and the unstable Muskat problem for the incompressible porous media equation (Saffman-Taylor instability). In both cases the fluid is smooth but at a curve where a hydrodynamic instability occurs. Experimentally, this instability triggers a laminar-turbulent transition in a neighborhood of the interface. Although unstable configurations in Hydrodynamics are very difficult to model, De Lellis-Székelyhidi’s version of convex integration have successfully describe several of these phenomena in the last years. Following this approach, we construct weak solutions for the two problems mentioned above. In the first one, we construct dissipative Euler flows for a large class of non-analytic vortex sheets without fixed sign. The mixed sign case was an open problem from the celebrated work of Delort. In the second one, we construct mixing flows after the Saffman-Taylor and smoothness breakdown. This is the first existence result for partially unstable data. Furthermore, we present a quantitative h-principle which shows that: Outside the “turbulence zone” these solutions are smooth and equal to a “subsolution”. Inside the turbulence zone these (infinitely many) solutions can behave wildly, but at a macroscopic scale they are almost indistinguishable from the subsolution.
July 6, 2022
Youngjoon Hong :
3 p.m. in 712 SEO
Abstract
As artificial intelligence makes progress, deep neural nets are being
applied to increasingly complex problem setups. In response to the
emerging difficulties of these new setups, deep learning research
explores new modeling tools to enhance the predictive power of neural
nets. Differential equations are among the new tools that are being
incorporated into deep neural net models in various ways. In particular,
neural networks and deep-learning have shown promise in speeding up
scientific simulations. For solving PDEs, the goal is often to produce
a model which can be used to quickly generate sample data for
statistical analysis; this is often applied in the inverse problem of
trying to determine a system's initial conditions, given later
observations. The result is now an exciting new research field known
as scientific machine learning, where techniques such as deep neural
networks and statistical learning are applied to classical problems
of applied mathematics. In this tutorial, our intention is to provide
an accessible introduction to recent developments in the field of
numerical solution of linear and nonlinear partial differential
equations using techniques from machine learning and artificial
intelligence.
Sept. 12, 2022
Liet Vo :
4 p.m. in 1227 SEO
Abstract
In this talk, I will discuss the two fully discrete
Chorin-type projection methods for the stochastic Stokes equations with
general multiplicative noise. The first scheme is the standard Chorin scheme
and the second one is a modified Chorin scheme which is designed by employing the
Helmholtz decomposition on the noise function at each time step to produce
a projected divergence-free noise and a ``pseudo pressure" after combining the original
pressure and the curl-free part of the decomposition.
An $O(k^\frac14)$ rate of convergence is proved for the standard Chorin scheme,
which is sharp but not optimal due to the use of general noise, where $k$ denotes the time mesh size.
On the other hand, an optimal convergence rate $O(k^\frac12)$ is established for the modified Chorin scheme.
The fully discrete finite element methods are formulated by discretizing both semi-discrete Chorin schemes in space by the standard finite element method.
Suboptimal order error estimates are derived for both fully discrete methods. It is proved that
all spatial error constants contain a growth factor $k^{-\frac12}$, where $k$ denotes the
time step size, which explains the deteriorating performance of the standard Chorin scheme
when $k\to 0$.
Sept. 19, 2022
Andrej Zlatos :
4 p.m. in 1227 SEO
Abstract
Bounded vorticity solutions to the 2D Euler equations on singular domains are typically not close to Lipschitz near boundary singularities, which makes their uniqueness a difficult open problem. I will present a general sufficient condition on the geometry of the domain that guarantees global uniqueness for all solutions initially constant near the boundary. This condition is only slightly more restrictive than exclusion of corners with angles greater than $\pi$ and, in particular, is satisfied by all convex domains. Its proof is based on showing that fluid particle trajectories for general bounded vorticity solutions cannot reach the boundary in finite time. The condition also turns out to be sharp in the latter sense: there are domains that come arbitrarily close to satisfying it and on which particle trajectories can reach the boundary in finite time. The above results also extend to positive vorticity solutions on fairly irregular domains that may even contain corners with angles greater than $\pi$.
Oct. 3, 2022
Le Chen :
4 p.m. in 1227 SEO
Abstract
In this talk, we will present a recent joint work with
Dr. Nicholas Eisenberg (arXiv:2209.04771). This paper deals with the long term
behavior of the solution to the nonlinear stochastic heat equation $\partial u
/\partial t - \frac{1}{2}\Delta u = b(u)\dot{W}$, where $b$ is assumed to be a
globally Lipschitz continuous function and the noise $\dot{W}$ is a centered and
spatially homogeneous Gaussian noise that is white in time. Using the moment
formulas obtained in Chen & Kim [10] and Chen & Huang [9], we
identify a set of conditions on the initial data, the correlation measure and
the weight function $\rho$, which will together guarantee the existence of an
invariant measure in the weighted space $L^2_\rho(\mathbb{R}^d)$. In particular, our
result includes the parabolic Anderson model (i.e., the case when $b(u)
= \lambda u$) starting from the Dirac delta measure.
Oct. 10, 2022
Colette Guillope :
4 p.m. in 1227 SEO
Abstract
We consider the initial value problem for the fractional
nonlinear Schrödinger equation with a damping term.
We study the global existence and the scattering of
solutions. Noticeably, for the focusing nonlinearity,
it is shown that for a large enough damping, the size
of the solution depends only on the initial data, and
the solution exists for all times in a space of
regular enough functions.
Oct. 17, 2022
David Padilla-Garza :
4 p.m. in 1227 SEO
Abstract
Nonlinear plate theory described the energy of an incompressible and inextensible thin elastic sheet. In this work, we show a general rigorous derivation of a generalization of such a model for non-euclidean plates with microheterogeneous structures. We also analyze the limiting energy in some examples and discover interesting and counter-intuitive phenomena.
Oct. 24, 2022
Neelima Sharma :
4 p.m. in 1227 SEO
Abstract
Mechanical stability is vital for the fitness and survival of animals and is a crucial aspect of robot design and control. Mechanisms that impart stability to animal bodies can be broadly classified into the innate mechanical response of the body or open loop control and feedback control. I will discuss two examples, finger contact and overall body stability, to show how the innate mechanical response of the body is tuned for stability by shaping the neural control and evolution of animal form. By studying the linearized dynamics of a finger's internal degrees of freedom pushing on a hard surface, we show that human fingers and other musculoskeletal linkages are intrinsically prone to buckling-type postural instabilities. Humans rely on a family of convex neural activation strategies of muscles so that the elastic response of our muscles can suppress the intrinsic instability, but by limiting maximal fingertip forces. In the study on whole-body lateral stability during locomotion in terrestrial animals, we find that the scaling of body aspect ratio with size is likely driven by the scale-dependent unevenness of the natural terrain. Thus, we identify morphological and control features that allow animals to perform stably and robustly in noisy environments by investigating their unstable and marginally stable behaviors.
Oct. 31, 2022
Xiang Wan :
4 p.m. in 1227 SEO
Abstract
In dealing with uniform stabilization of parabolic problems near an unstable equilibrium solution, the first critical step of what has become a standard strategy is to ascertain, if possible, Kalman's controllability condition of the projected finite dimensional unstable component. It was discovered about 15 years ago that this property is equivalent to establishing Unique Continuation Properties (UCPs) for the adjoint suitably over-determined eigen-problem.
In this talk, with focus on 2D-3D-Boussinesq systems (coupling the N-S with a diffusion equation), several UCPs of the adjoint systems are established to achieve the desired controllability results. These include the required UCPs for the localized interior as well as the localized boundary-based uniform stabilization of an unstable Boussinesq-system. We will also go through the idea of the proof, which follows the pointwise Carleman-type estimates approach.
Nov. 7, 2022
David Nicholls :
4 p.m. in 1227 SEO
Abstract
The rapid and robust simulation of linear waves interacting with
layered periodic media is a crucial capability in many areas of
scientific and engineering interest. High-Order Perturbation of
Surfaces (HOPS) algorithms are interfacial methods which recursively
estimate scattering quantities via perturbation in the interface shape
heights/slopes. For a single incidence wavelength such methods are the
most efficient available in the parameterized setting we consider
here. In this talk we describe a generalization of one of these HOPS
schemes by incorporating a further expansion in the wavelength about a
base configuration which constitutes an "Asymptotic Waveform
Evaluation" (AWE). We not only provide a detailed specification of the
algorithm, but also verify the scheme and point out its benefits and
shortcomings. With numerical experiments we show the remarkable
efficiency, fidelity, and high-order accuracy one can achieve with an
implementation of this algorithm.
Nov. 14, 2022
Shankar Venkataramani :
4 p.m. in 1227 SEO
Abstract
Patterns with a nearly periodic microstructure are ubiquitous. Oftentimes, a useful (reduced) description of these patterns is in terms of an underlying phase field. This phase field, however, is not directly observable, and indeed, for typical patterns with defects, there might not even be a single-valued phase field that describes the global pattern.
I will give an overview of some approaches to studying the dynamics of defects in nearly periodic stripe patterns using a phase field description. In particular, I will highlight the interplay between ideas from topology, signal processing, variational analysis, and numerical methods in our approach. This is based on joint work with Nick Ercolani, Alan Newell, and Amit Acharya.
Nov. 21, 2022
Cyril Imbert :
4 p.m. in 1227 SEO
Abstract
In this talk, we discuss regularity of solutions of the Landau-Coulomb equation. We will see that if they are radially symmetric, then they are regular away from the origin. Such a result is obtained through a local study of regularity à la De Giorgi. A key step is to derive suitable local estimates in the spirit of local energy estimates for the Navier-Stokes equations. This is a joint work with François Golse and Alexis Vasseur.
Feb. 6, 2023
Xia Chen :
4 p.m. in 1227 SEO
Abstract
Recently, a precise intermittency for the hyperbolic Anderson model
$$\partial^2_tu(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(x),$$
has been established in Itô-Skorohod regime. In this talk, we discuss the same problem in Stratonovich regime. Our approach provides new ingredient on representation and computation for Stratanovich moments.
The work is based on a collaborative project with Hu, Yaozhong.
Feb. 20, 2023
Yier Lin :
4 p.m. in 1227 SEO
Abstract
The Kardar-Parisi-Zhang (KPZ) equation is a nice model for random interface growth. In this talk, we will study the Freidlin–Wentzell LDP for the KPZ equation using the variational principle. Such an approach goes under the name of the weak noise theory in physics. We will explain how to extract various limits of the most probable shape of the KPZ equation in the setting of the Freidlin–Wentzell LDP. The talk is based on several joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.
March 6, 2023
Frederic Marazzato :
4 p.m. in 1227 SEO
Abstract
Origami folds have found a large range of applications in engineering as solar panels for satellites, or the fabrication of mechanical metamaterials. A homogenization process turning origami folds into smooth surfaces, developed in [Nassar et al, 2017], is first discussed. Then, its application to the eggbox pattern is presented alongside the PDEs characterizing the associated smooth surfaces. The talk will then focus on the PDEs describing Miura surfaces by studying existence and uniqueness of solutions and by proposing a numerical method to approximate them. Finally, some numerical examples are presented.
March 13, 2023
Pierre Yves Gaudreau Lamarre :
4 p.m. in 1227 SEO
Abstract
In quantum mechanics, the energy levels of microscopic physical systems (at the atomic or subatomic scale) correspond to the eigenvalues of a class of linear operators called Schrödinger operators. Therefore, one of the most fundamental problems in modern mathematical physics is to understand the spectrum of general Schrödinger operators.
In this talk, we are interested in Schrödinger operators perturbed by random noises. More specifically, we are interested in the rigidity of the spectrum of random Schrödinger operators; that is, the observation that the spectrum of some random operators has essentially the same structure before and after adding a random perturbation.
We will discuss a new approach to study this rigidity phenomenon that is based on the deep connections that exist between the spectral theory of Schrödinger operators and the solutions of parabolic PDEs. This talk is based on joint works with Promit Ghosal, Wenxuan Li and Yuchen Liao."
March 27, 2023
Xiaoan Shen :
4 p.m. in 1227 SEO
Abstract
We prove the existence of a class of orbitally stable bound state solutions to nonlinear Schrodinger equations with super-quadratic confinement in two and three spatial dimensions. These solutions are given by time-dependent rotations of a non-radially symmetric spatial profile which in itself is obtained via a doubly constrained energy minimization. One of the two constraints imposed is the total mass, while the other is given by the expectation value of the angular momentum around the z-axis. Our approach also allows for a new description of the set of minimizers subject to only a single mass constraint.
April 3, 2023
Gung-Min Gie :
4 p.m. in 1227 SEO
Abstract
Singular perturbations occur when a small coefficient
affects the highest order derivatives in a system of
partial differential equations. From the physical point
of view, singular perturbations generate thin layers near
the boundary of a domain, called boundary layers, where
many important physical phenomena occur. In this talk,
we discuss some recent results on the viscous boundary
layer analysis and their applications in implementing
effective numerical schemes including the Physics
Informed Neural Networks (PINNs).
April 10, 2023
Jinsil Lee :
4 p.m. in 1227 SEO
Abstract
In this talk, I will give a brief introduction to a trivariate spline collocation method for solving the Dirichlet problem of the 3D elliptic Monge-Ampère equation. Specifically, I will explain the spline collocation method introduced in [SIAM J. Numerical Analysis, 2405-2434,2022] to numerically solve iterative Poisson equations, and incorporate an averaged algorithm to ensure convergence of the iterations. Furthermore, I present various computational results, including testing known convex and non-convex solutions over convex and non-convex domains to demonstrate the efficiency and effectiveness of the method.
April 17, 2023
Matt Jacobs :
4 p.m. in 1227 SEO
Abstract
Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions?
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.
April 24, 2023
Lucas Bouck :
4 p.m. in 1227 SEO
Abstract
Liquid crystal polymeric networks are materials where a nematic liquid crystal is coupled with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Starting from the classical 3D trace formula energy of Bladon, Warner and Terentjev (1994), we derive a 2D membrane energy as the formal asymptotic limit of the 3D energy. The derivation is similar to derivations in Ozenda, Sonnet, and Virga (2020) and Cirak et. al. (2014). We characterize the zero energy deformations and prove that the energy lacks certain convexity properties. We propose a finite element method to discretize the problem. To address the lack of convexity of the membrane energy, we regularize with a term that mimics a higher order bending energy. We prove that minimizers of the discrete energy converge to minimizers of the continuous energy. For minimizing the discrete problem, we employ a nonlinear gradient flow scheme, which is energy stable. Additionally, we present computations showing the geometric effects that arise from liquid crystal defects. Computations of configurations from nonisometric origami are also presented.
Aug. 21, 2023
No Seminar :
4 p.m. in 636 SEO
Aug. 28, 2023
No Seminar :
4 p.m. in 636 SEO
Sept. 11, 2023
Mimi Dai :
4 p.m. in 636 SEO
Abstract
Derived two hundred years ago, the Navier-Stokes equation (NSE) governs the motion of fluids. In 1930s, Leray established the theory of weak solutions for the NSE and raised questions, some of which still remain open and center around the well-posedness problem. In the talk, we will review some progresses in the effort to understand these classical questions. The emphasis will be on some recent results, sparked by empirical laws in physics (such as Kolmogorov’s phenomenological theory of turbulence) and techniques from other fields in mathematics (for instance the convex integration scheme). We will also discuss some ongoing interests in various problems and new perspectives opened up by these techniques.
Sept. 18, 2023
Jiahong Wu :
4 p.m. in 636 SEO
Abstract
The Boussinesq equations concerned here model buoyancy-driven fluids such
as various atmospheric and oceanographic flows, and the Rayleigh-Benard
convection. This talk presents recent stability results on the Boussinesq
equations with partial dissipation near the Couette flow. We are able to prove
the nonlinear stability and large-time behavior results by exploiting the
enhanced dissipation created by a linear non-self-adjoint operator. This is a
joint work with Wen Deng and Ping Zhang. If time permits, we will briefly
mention stability results concerning the hydrostatic balance.
Sept. 25, 2023
Jared Bronski :
4 p.m. in 636 SEO
Abstract
\[
u_t + u u_x = \eta u_{xxx} + u_{xx} \qquad \lim_{x \rightarrow \mp \infty }u = \pm 1
\]
Originally proposed by Whitham as a model for the propagation of tidal bores.
It was shown by Bona and Schonbek that front type traveling wave solutions exist for all $\eta$, unique modulo translation, and are monotone for $|\eta|\leq \frac14$, and by Pego that such solutions are stable to small perturbations for the monotone case. We present a new stability criteria that does not require a smallness assumption on the difference between the initial data and the traveling wave, and which can be shown to hold in an open set of $\eta$ values that includes the monotone case. This condition involves the number of bound states of a certain Schr\”dinger operator constructed from the front solution. We will also discuss some rigorous numerical calculations that give intervals in $\eta$ where this spectral condition is guaranteed to hold. Joint work with Blake Barker, Vera Hur and Zhao Yang.
Oct. 2, 2023
Frederic Dias :
4 p.m. in 636 SEO
Abstract
The HIGHWAVE project (2019-2025) is primarily on wave breaking. The major
novelty during the first half of the project has been the experimental
campaign with a smart boulder. The data, obtained by altering the breaking
position of a wave impacting a vertical cliff, have demonstrated the
influence of wave-impact mode (aerated, breaking or sloshing) on the
displacement of clifftop boulders. This experimental campaign has shown
the range of wave focusing positions most conducive to boulder movement
and the range of displacement values we may expect in laboratory
experiments. The absence of multiple repeated tests and the inability to
fully quantify scaling effects mean that future work will firstly seek
to reliably extrapolate these laboratory boulder displacement
measurements to the real-world scale by quantifying pressure scaling
errors using large scale tests, carrying out a larger number of repeated
tests, and obtaining frictional similarity between prototype and
laboratory scales. Additionally, a comparison between the importance of
the wave-breaking position with the significant wave height and peak
wave period should be carried out.
At the beginning of the project we have been going back and forth between
what one would like to measure to better understand wave breaking and what
can be realistically measured in a hostile environment. The real
potential impact of our results will be the degree to which they are able
to describe real oceanic free surface profiles in a sea state where
breaking occurs. However, although we have been able to access a range
of data sets from wave buoys, ADCPs, radars, stereo vision, we have been
surprised to note the lack of reliability of measurements of breaking
wave events recorded in a given measurement time series or image as
measurements hit technical recording limits or generate artifacts in the
data. Indeed, we have come to realize that both the quality of the data
and the sophistication of data analysis of existing wave measurements
from sensors are based on decades-old technologies and methodologies and
are unsatisfactory for the detailed study of breaking waves.
Several theoretical results have been obtained as well: they range from
new limiting configurations for surface waves to new links between
superharmonic instability and wave breaking. Interesting results on wave
forecasting and on the effect of rain on waves will also be presented.
Acknowledgements: This work is funded by the European Research Council
(ERC) under the EU Horizon 2020 research and innovation programme (grant
agreement no. 833125-HIGHWAVE).
Oct. 16, 2023
Sayan Das :
4 p.m. in 636 SEO
Abstract
Sticky Brownian Motions are a family of correlated Brownian motions that have a tendency to stick together. They can be realized as the scaling limit of random walks in random environments and can also be viewed as random motions in a continuum-random environment. In this talk, I will present some results related to the quenched density of the motion of a particle in this continuum-random environment. In particular, I will show that under moderate deviation regime (a particular regime in between large deviation regime and diffusive regime), this quenched density after rescaling weakly converges to the solution of the stochastic heat equation with multiplicative space-time white noise. I will explain how our results shed light on the extreme behavior of Sticky Brownian Motions. This is a joint work with Hindy Drillick and Shalin Parekh.
Oct. 23, 2023
Jeffrey Larson :
4 p.m. in 636 SEO
Abstract
We present new methods for solving a broad class of bound-constrained smooth and
nonsmooth composite minimization problems. These methods are specially designed
for objectives that are some known mapping of outputs from a computationally
expensive function. We provide accompanying implementations of these methods: in
particular, a novel manifold sampling algorithm with subproblems that are in a
sense primal versions of the dual problems solved by previous manifold sampling
methods and a method that employs more difficult optimization subproblems. For
these two methods, we provide rigorous convergence analysis and guarantees. We
demonstrate extensive testing of these methods. Open-source implementations of
these methods are available.
Oct. 30, 2023
Fabrice Baudoin :
4 p.m. in 636 SEO
Abstract
We will give sufficient general conditions for the existence of Mosco limits of Korevaar-Schoen L2 energies, first in the context of Cheeger spaces and then in the context of fractal-like spaces with walk dimension greater than 2. Among the ingredients, a new Rellich- Kondrachov type theorem for Korevaar-Schoen-Sobolev spaces is of independent interest.
The talk will be based on a joint work with Patricia Alonso-Ruiz (Texas A&M)
Nov. 6, 2023
Soobin Cho :
4 p.m. in 636 SEO
Abstract
In this talk, I will explore efficient heat kernel estimates for jump-type Dirichlet forms and their associated Markov jump processes, specifically focusing on jump kernels that vanish at the boundary. In the first part, I will provide an overview of heat kernel estimates for jump processes in spaces with boundaries, encompassing actively reflected processes, killed processes, and censored processes. In the second part, I will present some new features of jump processes with boundary-vanishing jump kernels. The talk is based on joint work with Panki Kim (Seoul National University), Renming Song (UIUC) and Zoran Vondracek (University of Zagreb).
Nov. 13, 2023
Jue Yan :
4 p.m. in 636 SEO
Abstract
In this talk, we present the newly developed cell-average-based neural network (CANN). The method is based on the integral or weak formulation of the partial differential equations and is motivated by finite volume schemes. A “stencil” concept is introduced, and the network structure is designed to align with the conventional one-step methods. The well-trained network parameters are identified as the scheme coefficients of an explicit one-step method. The CANN method is found to be relieved from the small time step CFL restriction. A large time step can be used to evolve the solution forward in time explicitly. The method is remarkably efficient and fast and processes unique properties. The method allows for sharp evolution of contact discontinuity and shocks with almost zero numerical diffusion. The method can be generalized to solve out-of-distribution initial value problems accurately. Toward the end of the talk, we will discuss an ongoing collaborative project with Aerospace Engineering. Experimental data from the lab is used to train the CANN method and simulate the evolution of thin film water flow for the run-back water multi-phase problem.
Nov. 20, 2023
Matthew Novack :
4 p.m. in 636 SEO
Abstract
Compressible fluids are known to form shock waves, which can be represented by discontinuous solutions of the compressible Euler equations. However, physical shocks are actually continuous and in certain regimes can be represented by a smooth shock profile. In this talk, I will discuss a construction of weak shock profiles which solve the kinetic Landau equation. This is based on joint work with Dallas Albritton (Wisconsin) and Jacob Bedrossian (UCLA).
Nov. 27, 2023
Thuyen Dang :
4 p.m. in 636 SEO
Abstract
Topological insulators are materials that are insulating on the inside but are (electrically) conductive on their surface or edge. The conducting states are protected: in the presence of a defect, the transport on the edge is barely affected. This edge behavior is characterized by topological invariants of its quantization. It is known in the physics community that topological Anderson insulators (TAI) can be created by applying a disorder potential to the matter. The potential generates the phase transition of the matter that opens a spectral gap, which is the hallmark of topological insulators. In two dimensional, the TAI model can be recast as a Dirac equation. In this talk, we will discuss the formation of TAI from a mathematical homogenization point of view, and the connection between the microscopic and macroscopic Dirac operators. This is a joint work with Guillaume Bal.
Dec. 7, 2023
Frederic Weissler :
4 p.m. in 636 SEO
Abstract
After rapidly reviewing the history of the Korteweg-deVries equation, I will discuss the issues of local and global well-posedness for complex-valued
solutions on the circle. In particular, a large class of local-in time solutions develop a singularity in finite time. Hence global well-posedness can fail.
In addition, local well-posedness can fail for two reasons. There exist initial values with no reasonable local-in time solution. Also, if complex-valued solutions are
considered, continuous dependence fails at the zero solution. These results are of course completely at variance with the well-known results for real-valued solutions.
Jan. 8, 2024
No Seminar :
4 p.m. in 636 SEO
Jan. 22, 2024
No Seminar :
4 p.m. in 636 SEO
Feb. 5, 2024
Trevor Teolis :
4 p.m. in 636 SEO
Abstract
We present a new flocking model which has the versatility to capture the physically realistic qualitative behavior of the Motsch-Tadmor model, while also retaining the entropy law, which lends to a similar 1D global well-posedness analysis to the Cucker-Smale model. This is an improvement to the situation in the Cucker-Smale case, which may display the physically unrealistic behavior that large flocks overpower the dynamics of small, far away flocks; and it is an improvement in the situation in the Motsch-Tadmor case, where 1D global well-posedness is not known. We discuss the general well-posedness theory for the new model and the long-time behavior including alignment, strong flocking in 1D, and entropy estimates to estimate the distribution of the limiting flock, all of which extend the classical results of the Cucker-Smale case. In addition, we provide numerical evidence to show the similar qualitative behavior between the new model and the Motsch-Tadmor model.
Feb. 12, 2024
Trevor Leslie :
4 p.m. in 636 SEO
Abstract
The Euler Alignment system is a hydrodynamic PDE version of the celebrated Cucker-Smale ODE's of collective behavior. Together with Changhui Tan (University of South Carolina), we developed a theory of weak solutions in 1D, which provides a uniquely determined way to evolve the dynamics after a blowup. Inspired by Brenier and Grenier's work on the pressureless Euler equations, we show that the dynamics of interest are captured by a nonlocal scalar balance law, the unique entropy solution of which we generate through a discretization involving the "sticky particle Cucker-Smale" system. In this talk, we will discuss the formation of clusters of mass in the Euler Alignment system, and we will describe how to predict these clusters using the flux from the associated scalar balance law. The main results of this talk can be found in the preprint arXiv:2307.13626 (to appear in the Journal of Evolution Equations).
Feb. 26, 2024
Bingyu Zhang. :
4 p.m. in 636 SEO
Abstract
It is known that the solutions of the Cauchy problem for the Korteweg-de Vries (KdV) equation
$ u_t +uu_x +u_{xxx} =0, \quad u(x,0)= \phi (x), \quad x\in T, \ t\in R,$
and the viscous Burgers equation
$ u_t +uu_x - u_{xx} =0, \quad u(x,0)= \phi (x), \quad x\in T, \ t>0 $
posed on a periodic domain $T$, do not possess the sharp Kato smoothing property:
$ \phi \in H^s (T) \implies \partial ^{s+1}_xu \in L^{\infty}_x (T, L^2 (0,T))$.
Here, we discuss the equation,
$ u_t +uu_x +\alpha (x,t) u_{xxx} - \beta (x,t)u_{xx} =0, \qquad u(x,0)= \phi (x), \quad x\in T, \ t\geq 0, $
and demonstrate that if
$\int _{\mathbb{T}}\frac{\beta (x,t)}{|\alpha (x,t)|} dx >0 \quad \forall t\geq 0,$
and if it is locally well-posed in the space $ H^s (T)$ with $s \geq 0$,
then its solution $u$ possesses the sharp Kato smoothing property,
$ \phi \in H^s (T) \implies \partial ^{s+1}_xu \in L^{\infty}_x (T, L^2 (0,T)), \quad \forall \, s\geq 0. $
In addition, the nonlinear part of its solution $u$ possesses the strong Kato smoothing property,
$ \phi \in H^s (T) \implies (u -v)\in C([0,T]; H^{s+1} (T)), \quad \forall \, s>\frac12, $
and the double sharp Kato smoothing property
$ \phi \in H^s (T) \implies \partial ^{s+2}_x(u -v)\in L^{\infty}_x (\T, L^2 (0,T)), \quad \forall \, s>\frac12, $
with $v$ being the solution of the linear problem
$ v_t+ \alpha (x,t)v_{xxx} - \beta (x,t) v_{xx} =0, \quad v(x,0)=\phi (x), \quad x\in T, \ t>0. $
March 4, 2024
Colette Guillope :
4 p.m. in 636 SEO
Abstract
A second-order correct model for unidirectional water wave propagation in a channel is analysed. Earlier work showed this model to be well-posed in physically relevant regimes for auxiliary data in the Sobolev space $H^2$. The present theory shows global well-posedness in $H^1$. (This is joint work with J. Bona and H. Chen.)
March 11, 2024
Qirui Peng :
4 p.m. in 636 SEO
Abstract
We construct non-unique weak solutions $\theta \in C^0_t C^{0-}_x$ for forced surface quasi-geostrophic (SQG) equations. This is achieved through a convex integration scheme adapted to the sum-difference system of two distinct solutions. Without external forcing, non-unique weak solutions $\theta$ in space $C^0_t C^\alpha_x$ with $\alpha < -1/5$ were constructed by Buckmaster, Shkoller and Vicol (2019) and Isett and Ma (2021).
March 25, 2024
No Seminar :
4 p.m. in 636 SEO
April 8, 2024
Chengyang Shao :
4 p.m. in 636 SEO
Abstract
In this talk, we aim to introduce a new method to solve conjugacy problems involving 'small denominators’. Traditional proofs addressing the regularity loss caused by these 'small denominators' have heavily relied on Newtonian / Nash-Moser iterations. We aim to show that the regularity loss can be compensated by composing with appropriate para-differential operators, enabling a proof using only standard fixed-point schemes. This para-differential approach can be applied to study KAM-type problems, avoiding the commonly used 'KAM iterations.' The talk is based on joint works with Thomas Alazard.
April 15, 2024
Sung-Jin Oh :
4 p.m. in 636 SEO
Abstract
We prove the local wellposedness of the Cauchy problems for the electron magnetohydrodynamics equations (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity. (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, it is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. Our result significantly improves upon the straightforward adaptation of the classical work of Kenig–Ponce–Rolvung–Vega on the quasilinear ultrahyperbolic Schrödinger equations, as the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola–Metcalfe–Tataru in the case of elliptic principal term.
April 22, 2024
Erkan Nane :
4 p.m. in 636 SEO
Abstract
Partial differential equations and random fields have been used as successful models in various areas of applied mathematics, statistical mechanics, theoretical physics, theoretical neuroscience, theory of complex chemical reactions, fluid dynamics, hydrology,
cosmology, mathematical finance, and other scientific areas. In this talk I will consider non-linear space-time fractional (stochastic) heat type equations. These types of time fractional (stochastic) heat type equations are attractive models that can be used to model phenomenon with random effects with thermal memory.
I will review my most recent work on
(i) continuous time random walk limits;
(ii) heat type Cauchy problems with fractional time derivatives; and
(iii) stochastic fractional equations. In particular, I will talk about the asymptotic behavior of the solution with respect to time and a parameter $\lambda$; intermittency.
These results are our recent joint work with Jebessa B Mijena, Mohammud Foondun, Sunday Asogwa and Guerngar Ngartelbaye.
April 29, 2024
Aynur Bulut :
4 p.m. in 636 SEO
Abstract
In this talk, we give an overview of several recent results where quantitative estimates play a key role. In the first part of the talk, we discuss new convex integration constructions for fluid systems with external forcing. We will then discuss a novel application of these ideas to the surface quasi-geostrophic (SQG) equation. Moving forward with the theme of quantitative estimates, in the second part of the talk we will describe new bounds for the defocusing energy-supercritical Nonlinear Schrödinger equation (NLS) and use these to give a universal blow-up criteria which goes below the scaling invariant threshold. These results are in line with a recent breakthrough construction of finite-time blow-up solutions, and in particular give the first generic result distinguishing potential defocusing blow-up phenomena from many of the known examples of blow-up in the focusing setting. At the end of the talk, we will briefly describe applications to related models.
Sept. 9, 2024
Xiaoan Shen :
4 p.m. in 636 SEO
Abstract
We consider semiclassically scaled, weakly nonlinear Schrödinger equations with external confining potentials and additional angular-momentum rotation term. This type of model arises in the Gross-Pitaevskii theory of trapped, rotating quantum gases. We construct asymptotic solutions in the form of semiclassical wave-packets, which are concentrated in both space and in frequency around a classical Hamiltonian phase-space flow. The rotation term is thereby seen to alter this flow, but not the corresponding classical action.
Sept. 23, 2024
Joey Zou :
4 p.m. in 636 SEO
Abstract
We discuss ongoing work, joint with J. Wunsch and M. Yang, which concerns extending the Gutzwiller Trace Formula from the case of smooth potentials to the case of potentials with conormal singularities. In the smooth case, the formula expresses an eigenvalue-counting function of a Schrödinger operator as a sum of certain dynamical quantities over periodic Hamiltonian trajectories. In the conormal case, a consideration of a WKB ansatz for the Schrödinger propagator suggests the sum should incorporate dynamical information about Hamiltonian trajectories which reflect at the site of the singularity. We discuss the variational formulation required to make sense of the dynamics of such trajectories, as well as the further work needed to complete the proof. We also present an explicit example of such a potential whose eigenvalue asymptotics can be computed; such asymptotics show the presence of reflected dynamics when applied to the Gutzwiller Trace Formula.
Sept. 30, 2024
Yuzhe Zhu :
4 p.m. in 636 SEO
Abstract
The boundary regularization effect for hypoelliptic
kinetic equations is limited. The solution with the simplest
zero inflow boundary conditions exhibits at most Hölder continuity
near the singular set of the boundary. We will discuss recent
results on hypoelliptic regularity and explain the smoothness
properties of solutions in the presence of boundary conditions
in certain cases.
Oct. 7, 2024
John Albert :
4 p.m. in 636 SEO
Abstract
Dumas, Lannes, and Szeftel have introduced a family of equations designed to model waves in nonlinear optics which have less symmetry than solutions of the nonlinear Schrödinger equation, in that they are not axisymmetric about the axis of propagation. Their equations can include regularization terms which are present in some but not all spatial directions. We study the stability of standing-wave solutions of such regularized nonlinear Schrödinger equations, and find that the regularization can increase the range of nonlinearities for which standing waves are stable. This effect is similar to that seen in the Benjamin-Bona-Mahony regularization of the Korteweg-de Vries equation, but notably, the effect is present even when the regularization terms are not present in all directions.
Oct. 21, 2024
Chao Wu :
4 p.m. in 636 SEO
Abstract
We consider the electron magnetohydrodynamics (MHD) with static background ion flow. A special situation was studied numerically by physicists. In this paper we show the existence of determining wavenumber for the electron MHD, and establish a regularity condition only on the low modes of the solution.
Nov. 4, 2024
William Golding :
4 p.m. in 636 SEO
Abstract
The Landau equation in kinetic theory is one of the fundamental kinetic equations that describes the evolution of collisional plasmas. The equation includes a quadratic, non-local term that models the effects of binary collisions mediated by the Coulomb force. This collision term introduces substantial mathematical challenges, leaving many fundamental questions--such as the existence of global-in-time smooth solutions--largely open.
In this talk, I will explore recent progress made in understanding a simplified model, the homogeneous Landau equation, which retains the complex collision term. In a recent breakthrough work, Luis Silvestre and Nestor Guillen showed the existence of a new monotone functional---the Fisher information---which is used to construct global-in-time solutions for smooth rapidly decaying initial data. I will discuss joint work with Maria Gualdani and Amelie Loher, where we extend these results to general initial data and obtain new results on global-in-time existence and various forms of uniqueness. I will conclude with a discussion of how these results inform future research of the full model.
Nov. 25, 2024
Eduard Kirr :
4 p.m. in 636 SEO
Abstract
In an effort to find all bound states supported by the nonlinear Schrödinger equation we discovered that as their frequency approaches infinity each bound state separates into peaks which behave like particles. More precisely each peak localizes at a point in space and the force exerted by the potential depends solely on its position while the forces exerted by the other peaks depend on their relative positions. For the bound state to exist all peaks must be stationary hence the forces acting on it must sum to zero. Thus we get a system of algbraic equations which has unique solutions near a local minima of the potential but has infinitely many solutions near a saddle or local maxima. For the latter cases we employ a computer assisted proof to determine and classify a large set of solutions. This is joint work with A. Zarnescu (BCAM) and D. Manea (U. Bucharest).
Dec. 2, 2024
Konstantin Matetski :
4 p.m. in 636 SEO
Abstract
Polynuclear growth is one of the basic models in the Kardar-Parisi-Zhang universality class, which describes a one-dimensional crystal growth. For a particular initial state, its one-point value equals the length of the longest increasing subsequence for uniformly random permutations (whose asymptotic behavior was first studied by S. Ulam). In my joint work with J. Quastel and D. Remenik, we computed the distribution function of the polynuclear growth with arbitrary initial conditions. These formulas allowed us to express the distribution function in terms of the solutions of the Toda lattice, one of the classical integrable systems. A suitable rescaling of the model yields a non-trivial continuous limit of the polynuclear growth (the KPZ fixed point) and the respective equations (Kadomtsev-Petviashvili).
Jan. 27, 2025
Bartosz Protas :
4 p.m. in 636 SEO
Abstract
Rigorous mathematical analysis of the equations governing the
motion fluids leads to various a priori bounds expressing fundamental
limitations on the forms of extreme behavior possible in fluid flows.
In relation to turbulence, such bounds concern, for example, the
maximum production of enstrophy and the maximum energy or enstrophy
dissipation realizable in Navier-Stokes flows under different
constraints. While by virtue of how they are obtained such a priori
bounds account for all possible flow evolutions, they are often
conservative and hence amenable to improvement. We will present a
framework allowing one to systematically test the sharpness of such
bounds by solving a family of suitably-defined PDE optimization
problems. They are solved computationally using an adjoint-based
Riemannian gradient method. This approach will be illustrated with two
classical problems. First, we consider the question of (the absence of)
the dissipation anomaly in 2D Navier-Stokes flows. After recalling some
rigorous priori estimates describing the vanishing of the enstrophy
dissipation in the inviscid limit, we solve a family of PDE
optimization problems aimed at maximizing this quantity with respect to
the initial data. These results show that the extreme behavior found in
this way saturates an estimate due to Ciampa, Crippa & Spirito (2021),
thereby demonstrating the sharpness of this bound. The second problem
we discuss is motivated by the question about the possibility of
finite-time singularity formation in 3D Navier-Stokes flows. The
mathematical analysis of this problem revolves around conditional
regularity results which provide bounds that must be satisfied by all
smooth (classical) solutions, such that violation of these bounds
signals formation of a singularity. Our optimization-based approach
allows one to systematically search for the most singular behavior
possible in Navier-Stokes flows. However, no evidence for singularity
formation was detected in extreme flows realizing such worst-case
scenarios.
Joint work with D. Kang, P. Matharu, E. Ramirez and T. Yoneda
Feb. 3, 2025
Tianyu Zhu :
4 p.m. in 636 SEO
Abstract
Over the past twenty years, the field of plasmonics has been revolutionized with the isolation and utilization of two--dimensional materials, particularly graphene. Consequently there is significant interest in rapid, robust, and highly accurate computational schemes which can incorporate such materials. Standard volumetric approaches can be contemplated, but these require huge computational resources. Here we describe an algorithm which addresses this issue for nonlocal models of the electromagnetic response of graphene. Our methodology not only approximates the graphene layer with a surface current, but also reformulates the governing volumetric equations in terms of surface quantities using Dirichlet--Neumann Operators. We have recently shown how these surface equations can be numerically simulated in an efficient, stable, and accurate fashion using a High--Order Perturbation of Envelopes methodology. We extend these results to the nonlocal model mentioned above, and using an implementation of this algorithm, we study absorbance spectra of TM polarized plane--waves scattered by a periodic grid of graphene ribbons.
Feb. 10, 2025
Robert Neel :
4 p.m. in 636 SEO
Abstract
We first note that derivatives of the heat kernel, for small time, can be effectively localized. This allows us to extend well-known bounds on the logarithmic derivatives of the heat kernel from compact Riemannian manifolds to appropriate compact subsets of incomplete Riemannian, or even sub-Riemannian, manifolds. Moreover, for any pair of points in such a compact, we see that the asymptotics of the log-derivatives of the heat kernel are given by cumulants of geometrically natural random variables with respect to the law of large numbers measure of the corresponding Brownian bridge. This talk is based on joint work with Ludovic Sacchelli.
Feb. 17, 2025
Jeaheang Bang :
4 p.m. in 636 SEO
Abstract
Self-similar solutions play an important role in understanding the regularity and asymptotic behavior of solutions to the Navier-Stokes equations. We recently showed that axisymmetric self-similar solutions to the stationary Navier-Stokes equations in an $n$-dimensional cone with the no-slip boundary condition except at the origin must be trivial when $n\geq 4$. It rules out this particular scenario of boundary singularity, which has finite Dirichlet energy when $n\geq 5$. The main idea is to apply ODE techniques along with a sign property of the head pressure. This is a joint work with Changfeng Gui, Hao Liu, Yun Wang and Chunjing Xie.
Feb. 24, 2025
Perry Kleinhenz :
4 p.m. in 636 SEO
Abstract
For the damped wave equation on the torus, the energy decay rate is known to depend on the geometry of the support of the damping, and growth properties of the damping near where it is zero. In prior work, these growth properties are polynomial bounds on the damping, or derivative bound conditions, which control the gradient of the damping by a power of the damping. In this talk, we will show how these rates can be improved, and generalized to exponentially, or poly-logarithmically, growing damping, as well as more general non-polynomial derivative bound conditions. The proof of these results relies on resolvent estimates on very fine semiclassical scales. Time permitting, we will discuss how these new decay rates change when the geometry of the support of the damping changes.
March 10, 2025
Zhimeng Ouyang :
4 p.m. in 636 SEO
Abstract
Lattice models play a pivotal role in the investigation of microscopic multi-particle systems, with their continuum limits forming the foundation of macroscopic effective theory. These models have found wide-ranging applications in condensed matter physics, numerical analysis, and analysis of PDEs.
In this talk, I will present our recent work on the continuum limits of some lattice models to the corresponding nonlinear dispersive equations. Using the integrable Ablowitz–Ladik system as a prototype, we establish that solutions of this discrete model converge to solutions of either the cubic nonlinear Schrödinger equation (NLS) or the modified Korteweg–de Vries equation (mKdV) in certain limiting regimes. Notably, we consider white-noise-like initial data which excites Fourier modes throughout the circle, and demonstrate convergence to a system of NLS/mKdV. This result suggests that a sole continuum equation may not suffice to encapsulate the lattice dynamics in such a low-regularity setting akin to thermal equilibrium.
I will also outline the framework of our proof and discuss its broader implications, including its extension to more general lattice approximations of dispersive PDEs. In particular, our approach provides new insights into constructing dynamics for the Landau–Lifshitz spin model in its Gibbs state.
March 17, 2025
Johnny Yang :
4 p.m. in 636 SEO
Abstract
This talk explores stochastic partial differential equations (SPDEs) across different spatial dimensions. We will start by talking about SPDEs in one spatial dimension and some classical results. Then, we will discuss SPDEs on fractional spatial dimensions in [1,2) and discuss their solvability and various comparison principles. Finally, we will extend an analytical method to solve singular SPDEs on spatial dimensions in between two and three.
March 31, 2025
No Seminar :
4 p.m. in 636 SEO
April 7, 2025
Qin Li :
4 p.m. in 636 SEO
Abstract
In the past decade, there has been a significant shift in the types of mathematical objects under investigation, moving from vectors and matrices in Euclidean spaces, to functions residing in Hilbert or Banach spaces, and ultimately extending to probability measures within the probability measure space. Many questions that were originally posed in the context of linear function spaces are now being revisited in the realm of probability measures. One such question is to efficiently find a probability measure that minimizes a given objective functional. In Euclidean space, we devised optimization techniques like gradient descent and introduced momentum-based methods to accelerate the convergence. Now, the question arises: Can we employ analogous strategies to expedite convergence within the probability measure space?
We provide an affirmative answer to this question and show that momentum-based acceleration for Euclidean optimization now translates to Hamiltonian flows, and it can achieve arbitrary high-order of convergence. This opens the door of developing methods beyond standard gradient flow.
April 14, 2025
Jeonghun Lee :
4 p.m. in 636 SEO
Abstract
Poroelasticity equations arise from many applications in geophysics and biomechanics, so numerical simulations of poroelasticity equations are of great interest.
In this talk I discuss advanced finite element methods for poroelasticity and related problems.
In the first part, I introduce parameter-robust discretization of poroelasticity and explain that efficient preconditioners can be obtained by the operator preconditioning approach.
In the second part, I present hybridizable discontinuous Galerkin (HDG) methods for the problems that Stokes/Navier-Stokes equations and porous/poroelastic equations are coupled with interfaces.
The talk is based on joint works with K.-A. Mardal (University of Oslo), M. E. Rognes (Simula Research Laboratory), A. Cesmelioglu (Oakland University) S. Rhebergen (University of Waterloo), and other collaborators.
April 21, 2025
Maurice Fabien :
4 p.m. in 636 SEO
Abstract
We introduce a positivity-preserving discontinuous Galerkin (DG) scheme for hyperbolic PDEs on unstructured meshes in 2D and 3D. The standard DG spaces are augmented with either polynomial or non-polynomial basis functions. The primary purpose of these augmented basis functions is to ensure that the cell average from the unmodulated DG scheme remains positive. We explicitly obtain suitable basis functions by inspecting the method of characteristics on an auxiliary problem. A key result is proved which demonstrates that the augmented DG scheme will retain a positive cell average, provided that the inflow, source term, and variable coefficients are positive. Standard slope limiters can then be leveraged to produce a high-order conservative positivity-preserving DG scheme. Numerical experiments demonstrate the scheme is able to retain high-order accuracy as well as robustness for variable coefficients and nonlinear problems.
April 28, 2025
Razvan Radu :
4 p.m. in 636 SEO
Abstract
I will discuss the Nash iterative construction of non-conservative weak solutions to the Euler equations, with a particular focus on the difficulties presented by the two-dimensional case. I will then present a linear decoupling method, which enabled the construction of examples achieving sharp regularity for the 2D Euler equations, as well as for other systems.
May 5, 2025
Alexey Cheskidov :
noon in 636 SEO
Abstract
In the past couple of decades, mathematical fluid dynamics has
made significant strides with numerous constructions of solutions to fluid
equations that exhibit pathological or wild behaviors. These include the loss
of the energy balance, non-uniqueness, singularity formation, and dissipation
anomaly. Interesting from the mathematical point of view, providing
counterexamples to various well-posedness results in supercritical spaces,
such constructions are becoming more and more relevant from the physical
point of view as well. Indeed, a fundamental physical property of turbulent
flows is the existence of the energy cascade. Conjectured by Kolmogorov, it
has been observed both experimentally and numerically, but had been
difficult to produce analytically. In this talk I will overview new developments
in discovering not only pathological mathematically, but also physically
realistic solutions of fluid equations.
Sept. 8, 2025
Chushan Wang :
4 p.m. in 636 SEO
Abstract
The nonlinear Schrödinger equation (NLSE) arises from various applications in quantum physics and chemistry, nonlinear optics, plasma physics, Bose-Einstein Condensates, etc. In these applications, it is necessary to incorporate low-regularity or singularity into the NLSE, which may arise from the potential, nonlinearity, and/or initial data. Typical examples include the discontinuous square-well potential, the singular Coulomb potential, the non-integer power nonlinearity, the logarithmic nonlinearity, and initial data that are ground states of the Schrödinger operator with such potential. Such low regularity and singularity pose significant challenges in the analysis of standard numerical methods and the development of novel accurate, efficient, and structure-preserving numerical schemes.
In this talk, I will introduce several new analysis techniques to establish optimal error bounds for some widely used numerical methods under optimally weak regularity assumptions. Based on the analysis, we also propose novel temporal and spatial discretizations to handle the low regularity and singularity more effectively.
Sept. 29, 2025
Hassan Babaei :
4 p.m. in 636 SEO
Abstract
In this talk, I will present the construction of solutions to the one-dimensional Dirac
equation on the half-line with Dirichlet boundary conditions. While the Dirac equation is a
four-dimensional system arising in quantum field theory, I will focus on the one-
dimensional initial-boundary value problem. The primary analytical tool is the unified
transform method (or known as Fokas method), which provides an explicit representation
of the solution. To introduce the method, I will first demonstrate it in the context of the heat
equation on the half-line. I will then apply it to the Dirac equation to derive explicit solution
formulas and analyze the associated boundary behavior at the origin. Furthermore, I will
discuss the long-time dynamics of these solutions. If time permits, I will conclude with a
discussion of Sobolev-space energy estimates for the solutions, including control of both
spatial norms and time-regularity.
Oct. 6, 2025
Jeremy Hoskins :
4 p.m. in 636 SEO
Abstract
Flexural waves, the propagation of waves in thin elastic sheets, arise in a number of contexts, and, particularly, in the study of ice shelves. In the frequency domain, they are commonly modeled as a fourth order PDE in two dimensions with clamped plate, free plate, or supported plate boundary conditions. Here, we review existing approaches for solving boundary value problems of this type, and discuss some limitations. Building on this, for the supported plate and free plate problems, we propose novel representations which ultimately reduce the problems to second kind integral equations. Moreover, the resulting integral equations are amenable to standard high order discretization approaches and fast algorithms. Several numerical examples will be presented which illustrate the properties of these integral equations. Finally, generalizations to other wave phenomena will be discussed.
Oct. 13, 2025
Jiajie Chen :
4 p.m. in 636 SEO
Abstract
In this talk, we will present recent developments in constructing self-similar singularities in the compressible Euler equations and the nonlinear wave equation, associated with implosion. Our approach combines ODE techniques, weighted energy estimates, compact perturbation methods, and soft functional analysis arguments.
Oct. 20, 2025
Anjali Nair :
4 p.m. in 636 SEO
Abstract
A well-known conjecture in physical literature states that high frequency waves propagating over long distances through turbulence eventually become complex Gaussian distributed. The intensity of such wave fields then follows an exponential law, consistent with speckle formation observed in physical experiments. Though fairly well-accepted and intuitive, this conjecture is not entirely supported by any detailed mathematical derivation. In this talk, I will discuss some recent results demonstrating the Gaussian conjecture in a weak-coupling regime of the paraxial approximation.
The paraxial approximation is a high frequency approximation of the Helmholtz equation, where backscattering is ignored. This takes the form of a Schrödinger equation with a random potential and is often used to model laser propagation through turbulence. In particular, I will describe a diffusive scaling where the limiting probability distribution of the wavefield is completely described by a second moment which follows an anomalous diffusion. The proof relies on the asymptotic closeness of statistical moments of the wavefield under the paraxial approximation, its white noise limit and the complex Gaussian distribution itself. An additional stochastic continuity/tightness criterion allows to show the convergence of these distributions over spaces of Hölder-continuous functions. Numerical simulations illustrate theoretical results.
This is joint work with Guillaume Bal.
Nov. 3, 2025
David Nicholls :
4 p.m. in 636 SEO
Abstract
In many applications of scientific and engineering interest, the
accurate modeling of linear waves scattered by quasiperiodic media
plays a crucial role. The ability to numerically simulate such
configurations robustly and rapidly is of overwhelming importance in
photonics applications. In this talk we will discuss the specific
problem of electromagnetic radiation interacting with a
two-dimensional multiply layered diffraction grating with
quasiperiodic interfaces. We describe how the classical boundary
perturbation method of Field Expansions can be extended to this
two-dimensional problem, and with specific numerical experiments we
will show the remarkable efficiency, fidelity, and high-order accuracy
one can achieve with an implementation of this algorithm.
Nov. 24, 2025
Hongyi Chen :
4 p.m. in 636 SEO
Abstract
We identify conditions for which a Dirichlet space (a metric measure space with diffusion) admitting a sub-Gaussian heat kernel would be in the Da Prato-Debussche regime of the $\Phi^{n+1}$ equation. For this purpose, we use heat kernel based Besov spaces, where regularity of Schwartz-type distributions is measured using the small time behavior of the heat kernel. In the process, we show how many nontrivial parts of the solution theory such as construction of paraproducts and energy estimates are made more difficult by the roughness of the underlying geometry. These difficulties in fact produce a more restrictive regime than one may first expect by typical scaling heuristics.
Feb. 2, 2026
Dallas Albritton :
4 p.m. in 636 SEO
Abstract
The forced 2D Euler equations exhibit non-unique solutions with vorticity in L^p, p > 1, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of "resolving" the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity and consider epsilon-size perturbations of his initial datum. We discover a uniqueness threshold below which the vanishing viscosity solution is unique and radial, and at which certain vanishing viscosity solutions converge to non-unique, non-radial solutions. Joint work with Maria Colombo and Giulia Mescolini (EPFL).
Feb. 23, 2026
Linfeng Li :
4 p.m. in Zoom
Abstract
Quantitative unique continuation is a fundamental property of partial differential equations, which asserts that the smallness of a solution at one scale quantitatively controls its behavior at larger scales. It plays an important role in the study of nodal sets and in problems from control theory. While Yau's conjecture on nodal set remains open in full generality, there has been significant progress in recent years. In this talk, I will present several new results on the nodal sets of elliptic and parabolic equations, which extend Yau’s conjecture to solutions with Gevrey regularity. Subsequently, I will explore applications of quantitative unique continuation to observability inequalities in control theory, which assert that global behavior can be determined from partial information.
March 9, 2026
Francis Aznaran :
4 p.m. in 636 SEO
Abstract
For the discretisation of symmetric, divergence-conforming stress tensors in continuum mechanics, we prove inf-sup stability bounds which are uniform in polynomial degree and mesh size for the Hu–Zhang finite element in two dimensions. This is achieved via an explicit construction of a bounded right inverse of the divergence operator, with the crucial component being the construction of bounded Poincaré operators for the stress elasticity complex which are polynomial-preserving, in the Bernstein–Gelfand–Gelfand framework of the finite element exterior calculus. We also construct hp-bounded projection operators satisfying a commuting diagram property and hp-stable Hodge decompositions. Numerical examples are provided.
March 16, 2026
Michael Novack :
4 p.m. in 636 SEO
Abstract
We consider a geometric variational problem which models films/foams not as surfaces, but as regions with small but positive volume. In order to explain physical properties of films/foams called Plateau borders, we will present a partial regularity theorem which differs from the classical theory in two key aspects. We will discuss these differences and also connections to free boundary problems such as spectral minimal partitions and the two-phase Bernoulli problem.
April 13, 2026
Gerard Awanou :
4 p.m. in 636 SEO
Abstract
The second boundary value problem for the Monge-Ampere equation is central to applications in illumination
design, such as the construction of refractors and reflectors. While semi-discrete optimal transport methods
have worst-case computational complexity of O(N^2) in dimensions 2 and 3, finite difference methods have
linear complexity O(N) when used with a stencil of size independent of the number of mesh points N.
This talk will present a complete theoretical foundation—covering existence, uniqueness, and convergence—for
a linear-complexity finite-difference discretization based on a reformulation of the second
boundary condition that prescribes the asymptotic cone of the epigraph of a convex extension of the solution.
April 27, 2026
Young-Pil Choi :
4 p.m. in 636 SEO
Abstract
We study a continuum Lagrangian alignment system for interacting agents with weak initial data. We first prove global well-posedness of the Lagrangian dynamics and derive quantitative flocking estimates. We then pass from the Lagrangian description to an Eulerian one, and obtain an Euler-Reynolds-alignment system involving a nonnegative Reynolds stress and, in the nonlinear velocity-coupling case, an additional defect force caused by microscopic velocity fluctuations. Under a heavy-tailed interaction assumption, we show that these defect terms vanish asymptotically, leading to mono-kinetic closure at large times. In the linear velocity-coupling case, we further prove the global existence of weak solutions to the Euler-alignment system, including a sharp critical-threshold result in one dimension and a global existence result in higher dimensions under a large-coupling condition. We also establish mean-field convergence results for the underlying particle system, including uniform-in-time convergence in the linear case.