Convergence of the Euler-Voigt equations to the Euler equations in two dimensions
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910854582108160 |
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| author | Abbate, Stefano Berselli, Luigi C. Crippa, Gianluca Spirito, Stefano |
| author_facet | Abbate, Stefano Berselli, Luigi C. Crippa, Gianluca Spirito, Stefano |
| contents | 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 $C([0,T];L^2(\mathbb{T}^2))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01270 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of the Euler-Voigt equations to the Euler equations in two dimensions Abbate, Stefano Berselli, Luigi C. Crippa, Gianluca Spirito, Stefano Analysis of PDEs 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 $C([0,T];L^2(\mathbb{T}^2))$. |
| title | Convergence of the Euler-Voigt equations to the Euler equations in two dimensions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.01270 |