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

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.