Ill-posedness and inviscid limit of basic equations of fluid dynamics in Besov spaces

TL;DR

This paper investigates the ill-posedness and inviscid limit failure of fundamental fluid dynamics equations in Besov spaces, providing new initial data constructions and simplified proofs for these phenomena.

Contribution

It introduces a new initial data construction and simplifies proofs of ill-posedness and inviscid limit failure in Besov spaces for Euler, Navier-Stokes, and surface quasi-geostrophic equations.

Findings

Proves ill-posedness of solutions in $B^s_{p, abla}$ spaces.

Shows failure of $B^s_{p, abla}$-convergence in the inviscid limit.

Extends results to both Euler and surface quasi-geostrophic equations.

Abstract

In this paper, we consider the Cauchy problem to the basic equations of fluid dynamics on the torus. Firstly, we construct a new initial data and provide a simple proof on the ill-posedness of $B^s_{p,\infty}$ solution of the Euler equations and the surface quasi-geostrophic equation, which covers the results obtained by Cheskidov-Shvydkoy \cite{CS} and Misio{\l}ek-Yoneda \cite{MY}. Secondly, we prove the failure of the $B^s_{p,\infty}$-convergence in the inviscid limit for both the Navier-Stokes equations and the surface quasi-geostrophic equation.