The Nonlocal-to-Local Limit for the Inviscid Leray-{\alpha} Equations

TL;DR

This paper proves that solutions of the inviscid Leray-α equations converge to Euler solutions in certain Sobolev spaces and also establishes convergence of weak solutions under specific turbulence-inspired scaling assumptions.

Contribution

It demonstrates the nonlocal-to-local limit for inviscid Leray-α equations, connecting nonlocal regularizations to classical Euler solutions in both strong and weak senses.

Findings

Strong solutions of Leray-α converge to Euler solutions in Sobolev spaces.

Weak solutions under turbulence-inspired scaling converge to Euler solutions in L^2.

The results hold for a broad class of regularising kernels.

Abstract

We consider the inviscid Leray-$\alpha$ equations - an inviscid nonlocal regularisation of the Euler equations. In the first part, we prove the convergence of strong solutions of the Leray-$\alpha$ equations to strong solutions of the Euler equations in $H^s(\mathbb{R}^d)$ for $s>d/2 +1 $, $d\in \{2,3\}$, for a large class of regularising kernels. In the second part, we consider weak solutions on a bounded domain with a local scaling property far away from the boundary. The scaling relates to second-order structure functions from turbulence theory and does not imply regularity. Nonetheless, under these assumptions, the weak solutions converge to (possibly wild) weak solutions of Euler in $L^2$ for almost every $t$.