Bingyu Zhang. : On Some Joint Effects of Dispersion and Dissipation of a Class of Nonlinear Evolution Equations

Posted by Jerry Bona , part of the Analysis and Applied Mathematics Seminar.

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Feb. 26, 2024, 4 p.m.

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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. $