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

Abstract

We consider the limit $\alpha \to 0$ 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.

References in corpus (2)

• Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: indifference to boundary layers
• Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions