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The Nonlocal-to-Local Limit for the Inviscid Leray-α Equations

Jule Schindler, Emil Wiedemann

TL;DR

This work analyzes the nonlocal-to-local (α→0) limit of the inviscid Leray-$α$ equations, addressing both strong convergence on $\mathbb{R}^d$ and weak convergence on bounded domains. It proves strong convergence in $H^s$ for $s> d/2+1$ and $d\in\{2,3\}$ for a broad kernel class, with explicit rates in $H^{s'}$, and it shows weak convergence to Euler solutions on bounded domains under a turbulence-inspired local scaling property, even allowing for anomalous dissipation. The results are obtained without relying on vorticity and under general regularising kernels $K^{α}$, clarifying the relationship between Leray-$α$ regularisation and the Euler dynamics. These findings have implications for understanding nonlocal regularisations in fluid dynamics and for turbulence modeling via α-regularisation approaches.

Abstract

We consider the inviscid Leray-$α$ equations - an inviscid nonlocal regularisation of the Euler equations. In the first part, we prove the convergence of strong solutions of the Leray-$α$ 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 $t$.

The Nonlocal-to-Local Limit for the Inviscid Leray-α Equations

TL;DR

This work analyzes the nonlocal-to-local (α→0) limit of the inviscid Leray- equations, addressing both strong convergence on and weak convergence on bounded domains. It proves strong convergence in for and for a broad kernel class, with explicit rates in , and it shows weak convergence to Euler solutions on bounded domains under a turbulence-inspired local scaling property, even allowing for anomalous dissipation. The results are obtained without relying on vorticity and under general regularising kernels , clarifying the relationship between Leray- regularisation and the Euler dynamics. These findings have implications for understanding nonlocal regularisations in fluid dynamics and for turbulence modeling via α-regularisation approaches.

Abstract

We consider the inviscid Leray- equations - an inviscid nonlocal regularisation of the Euler equations. In the first part, we prove the convergence of strong solutions of the Leray- equations to strong solutions of the Euler equations in for , , 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 for almost every .
Paper Structure (4 sections, 8 theorems, 97 equations)

This paper contains 4 sections, 8 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convergence of strong solutions in $H^s$
  4. Convergence of weak solutions with local scaling property

Key Result

Theorem 1

Let Then, for all $0<T<T^*$, there exists $0<\bar{\alpha} = \bar{\alpha}(v_0,v_{0}^{\alpha}, T)$ such that for all $\alpha \leq \bar{\alpha}$ there is a unique solution $v^{\alpha} \in \mathcal{C}([0,T];H^s_{}) \cap W^{1,1}(0,T;H^{s-1}_{})$ of the Leray-$\alpha$ equations with initial datum $v^{\alpha}_ and for all $0\leq t \leq T$, $s' \in [0,s-1]$, where $C=C(\|v_0\|_{H^s},T)$ is independent o

Theorems & Definitions (16)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Lemma 1
  • Proposition 1: Existence of strong solutions for Leray-$\alpha$
  • Proposition 2: Existence of strong solutions for Euler (e.g. Bedrossian.2022Kato.1972Majda.2010)
  • Remark 3
  • Remark 4
  • ...and 6 more