The Burgers' Equation in the Hermite-Sobolev Spaces

Abstract

In this paper, we show existence and uniqueness of solutions to the viscous Burgers' equation in ${\mathbb{R}}^d$, when the initial condition $u_0$ is in Hermite-Sobolev space of index $p$, for suitable non-negative integers $p$. Our solutions are local in time. We also have a regularity result, viz. if $u_0$ belongs to Schwartz space, then so does the solution.