Non-uniform dependence for Euler equations in Besov spaces

TL;DR

This paper demonstrates that the solution map for incompressible Euler equations is not uniformly continuous in certain Besov spaces, highlighting limitations in the stability of solutions.

Contribution

It establishes the non-uniform dependence of solutions on initial data in Besov spaces for the Euler equations, under conditions ensuring local existence and uniqueness.

Findings

Solution map is not uniformly continuous in specified Besov spaces.

Results apply to parameters where local existence and uniqueness are valid.

Highlights limitations in stability and continuous dependence for Euler solutions.

Abstract

We prove the non-uniform continuity of the data-to-solution map of the incompressible Euler equations in Besov spaces $B_{p,q}^{s}$, where the parameters $p, q$ and $s$ considered here are such that the local existence and uniqueness result holds.