Uniform time of existence for the alpha Euler equations

TL;DR

This paper proves uniform local existence of strong solutions for the alpha-Euler equations on bounded domains, and demonstrates their convergence to classical Euler solutions as alpha approaches zero, using new a priori estimates.

Contribution

It establishes uniform existence results for alpha-Euler equations and their convergence to Euler equations, along with new local existence results for Euler equations on bounded domains.

Findings

Uniform existence of strong solutions for alpha-Euler equations for small alpha.

Convergence of alpha-Euler solutions to Euler solutions as alpha approaches zero.

New a priori estimates in conormal spaces for these equations.

Abstract

We consider the $\alpha$-Euler equations on a bounded three-dimensional domain with frictionless Navier boundary conditions. Our main result is the existence of a strong solution on a positive time interval, uniform in $\alpha$, for $\alpha$ sufficiently small. Combined with the convergence result in a previous article by the same authors, this implies convergence of solutions of the $\alpha$-Euler equations to solutions of the incompressible Euler equations when $\alpha \to 0$. In addition, we obtain a new result on local existence of strong solutions for the incompressible Euler equations on bounded three-dimensional domains. The proofs are based on new {\it a priori} estimates in conormal spaces.