Existence of solutions to Burgers equations in domains that can be transformed into rectangles

Abstract

This work is concerned with Burgers equation $\partial _{t}u+u\partial_x u-\partial _x^2u=f$ (with Dirichlet boundary conditions) in the non rectangular domain $\Omega =\{(t,x)\in R^2;\ 0<t<T,\; \varphi_1(t)<x<\varphi _2(t)\}$ (where $\varphi _1(t)<\varphi _2(t)$ for all $t\in [ 0;T]$). This domain will be transformed into a rectangle by a regular change of variables. The right-hand side lies in the Lebesgue space $L^2(\Omega )$, and the initial condition is in the usual Sobolev space $H_0^{1}$. Our goal is to establish the existence, uniqueness and the optimal regularity of the solution in the anisotropic Sobolev space.

Keywords

• Semilinear parabolic problem
• Burgers equation
• existence
• uniqueness
• anisotropic Sobolev space