The Euler Equations in planar nonsmooth convex domains

Abstract

We consider the Euler system set on a bounded convex planar domain, endowed with impermeability boundary conditions. This system is a model for the barotropic mode of the Primitive Equations on a rectangular domain. We show the existence of weak solutions with L^p vorticity for 4/3<= p <= 2, extending and enriching a previous result of Taylor. In the physically interesting case of a rectangular domain, a similar result holds for all 2<p<\infty as well. Moreover, we show the uniqueness of solutions with bounded initial vorticity. The main tool is a new BMO regularity estimate for the Dirichlet problem on domains with corners.

Final version incorporating referee's suggestions

References in corpus (3)

• Mathematics and Turbulence: where do we stand?
• Dissipative continuous Euler flows in two and three dimensions
• Uniqueness for two dimensional incompressible ideal flow on singular domains

Cited by in corpus (8)

• Finite-time Singularity Formation for Strong Solutions to the Boussinesq System
• The Two Dimensional Euler Equations on Singular Exterior Domains
• The Euler Equations in Planar Domains with Corners
• Grisvard's shift theorem near L^infinity and Yudovich theory on polygonal domains
• Uniqueness for the 2-D Euler equations on domains with corners
• Very weak solutions of the Stokes problem in a convex polygon
• Euler Equations on General Planar Domains
• The 2D Euler-Boussinesq equations in planar polygonal domains with Yudovich's type data