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This is a Quantitative Study study.

October 13, 2025Open Access

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

Key Points

Weak solutions of the Euler equations exhibit strong convergence, improving understanding of fluid dynamics.
The rate of convergence for the unique Yudovich solution in L² is provided, offering precision in approximation.
Convergence rates for classical solutions are established, highlighting the role of regularity in two-dimensional spaces.
The study leverages Sobolev spaces to explore convergence in fluid motion, enhancing theoretical foundations in applied mathematics.

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;L² (T²) ) the approximating velocity converges strongly in C (0, T;H¹ (T²) ). Moreover, for the unique Yudovich solution of the 2D Euler equations we provide a rate of convergence for the velocity in C (0, T;L² (T²) ). 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;L² (T²) ).

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

Key Points

Abstract

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