Ill-posedness for the Euler equations in Besov spaces

TL;DR

This paper demonstrates ill-posedness of the Euler equations in certain Besov spaces by constructing initial data that leads to discontinuous solution maps at time zero, extending previous periodic results.

Contribution

It constructs explicit initial data in Besov spaces showing discontinuity of the solution map, thus establishing ill-posedness in these function spaces.

Findings

Solution map discontinuous at t=0 in specified Besov spaces

Extends previous periodic ill-posedness results to non-periodic case

Provides new insights into the regularity requirements for well-posedness

Abstract

In the paper, we consider the Cauchy problem to the Euler equations in $\mathbb{R}^d$ with $d\geq2$. We construct an initial data $u_0\in B^\sigma_{p,\infty}$ showing that the corresponding solution map of the Euler equations starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^\sigma_{p,\infty}$, which implies the ill-posedness for this equation in $B^\sigma_{p,\infty}$. We generalize the periodic result of Cheskidov and Shvydkoy \cite{Cheskidov}.