Weak solutions for the $\alpha$-Euler equations and convergence to Euler

TL;DR

This paper investigates the limit of the $\

Contribution

It proves the convergence of $\

Findings

Existence of global solutions for bounded vorticity.

Convergence to Euler solutions as $\

paper_type

Abstract

We consider the limit $\alpha\to0$ for the $\alpha$-Euler equations in a two-dimensional bounded domain with Dirichlet boundary conditions. Assuming that the vorticity is bounded in $L^p$, we prove the existence of a global solution and we show the convergence towards a solution of the incompressible Euler equation with $L^p$ vorticity. The domain can be multiply-connected. We also discuss the case of the second grade fluid when both $\alpha$ and $\nu$ go to 0.