Stefano Abbate | Searcharxiv

arXiv2025

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

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 $C([0,T];L^2(\mathbb{T}^2))$.

math.AP↗

arXiv2023

Strong convergence of the vorticity and conservation of the energy for the $α$-Euler equations

In this paper, we study the convergence of solutions of the $α$-Euler equations to solutions of the Euler equations on the $2$-dimensional torus. In particular, given an initial vorticity $ω_0$ in $L^p_x$ for $p \in (1,\infty)$, we prove strong convergence in $L^\infty_tL^p_x$ of the vorticities $q^α$, solutions of the $α$-Euler equations, towards a Lagrangian and energy-conserving solution of the Euler equations. Furthermore, if we consider solutions with bounded initial vorticity, we prove a quantitative rate of convergence of $q^α$ to $ω$ in $L^p$, for $p \in (1, \infty)$.

math.AP↗