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.

Fixed some typos. Fixed an error in the proof of Proposition 3.1. Added a short section on the Euler equations in $C^k$ and other systems

References in corpus (3)

• Strong illposedness of the incompressible Euler equation in integer $C^m$ spaces
• Nonlinear wave interactions for the Benjamin-Ono equation
• Ill-posedness of the incompressible Euler equations in the $C^1$ space

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