Convergence of the Euler-Voigt equations to the Euler equations in two dimensions

TL;DR

The paper analyzes the vanishing-$\alpha$ limit of the Euler-Voigt equations in 2D on the torus, proving convergence to the Euler equations under multiple regularity frameworks. It develops a compactness-based argument to obtain strong $C([0,T];H^1)$ convergence for weak Euler solutions and $C([0,T];L^2)$ convergence of vorticities, and extends to quantitative rates in the Yudovich class and for high Sobolev regularity, where explicit rates are given for both velocity and vorticity. The results hinge on uniform-in-$\alpha$ a priori estimates and careful treatment of the modified vorticity equation, leveraging Osgood-type stability and Grönwall inequalities. Overall, the work clarifies the consistency of Euler-Voigt as a surrogate for Euler in 2D and provides practical convergence rates relevant for simulations and LES-style modeling.

Abstract

In this paper, we consider the two-dimensional torus and we study the convergence of solutions of the Euler-Voigt equations to solutions of the Euler equations, under several regularity settings. More precisely, we first prove that for weak solutions of the Euler equations with vorticity in $C([0,T];L^2(\mathbb{T}^2))$ the approximating velocity converges strongly in $C([0,T];H^1(\mathbb{T}^2))$. Moreover, for the unique Yudovich solution of the $2D$ Euler equations we provide a rate of convergence for the velocity in $C([0,T];L^2(\mathbb{T}^2))$. Finally, for classical solutions in higher-order Sobolev spaces we prove the convergence with explicit rates of both the approximating velocity and the approximating vorticity in $C([0,T];L^2(\mathbb{T}^2))$.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Definition 2.2
• Proposition 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• proof