Ill-posedness for the Burgers equation in Sobolev spaces

TL;DR

This paper demonstrates that the Cauchy problem for the Burgers equation is ill-posed in Sobolev spaces with regularity index s between 1 and 1.5, highlighting limitations of well-posedness in these function spaces.

Contribution

The paper establishes the ill-posedness of the Burgers equation in Sobolev spaces for a specific range of regularity, extending understanding of solution behavior.

Findings

Ill-posedness in Sobolev spaces $H^s$ for $s o 1.5$

Limitations on solution regularity for the Burgers equation

Clarification of the boundary between well-posedness and ill-posedness

Abstract

In this paper, we considered the Cauchy problem for the Burgers equation and proved that the problem is ill-posed in Sobolev spaces $H^s$ with $s\in[1,\fr32)$.