Ill-posedness results in critical spaces for some equations arising in hydrodynamics

TL;DR

This paper introduces a new approach to studying ill-posedness in critical spaces for various hydrodynamic equations, demonstrating strong ill-posedness results for several important models.

Contribution

It develops a novel method for analyzing norm inflation and proves strong ill-posedness in critical spaces for multiple hydrodynamic equations.

Findings

Proves strong ill-posedness of Euler equations in certain critical spaces.

Applies the method to Oldroyd B, SQG, and Boussinesq systems.

Establishes norm inflation results in critical spaces for these equations.

Abstract

Many questions related to well-posedness/ill-posedness in critical spaces for hydrodynamic equations have been open for many years. In this article we give a new approach to studying norm inflation (in some critical spaces) for a wide class of equations arising in hydrodynamics. As an application, we prove strong ill-posedness of the $n$-dimensional Euler equations in the class $C^1\cap L^2 (\Omega)$ and also in $C^k \cap L^2(\Omega)$ where $\Omega$ can be the whole space, a smooth bounded domain, or the torus. We also apply our method to the Oldroyd B, surface quasi-geostrophic, and Boussinesq systems.