The 2D Euler equations are well-posed for generic initial data in $L^2$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915938971942912 |
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| author | Galeati, Lucio |
| author_facet | Galeati, Lucio |
| contents | In this note we show the existence of a residual set (in the sense of Baire) of divergence free initial data $u_0\in L^2(D)$, $D=\mathbb{R}^2$ or $\mathbb{T}^2$, for which global existence and uniqueness of weak solutions to the incompressible 2D Euler equations holds. The associated solutions $u$ satisfy the energy balance and are recovered in the vanishing viscosity limit from solutions to 2D Navier-Stokes, which as a consequence cannot display anomalous dissipation of energy. Additionally, there exists a unique regular Lagrangian flow associated to such $u$, and the associated transport equation is well-posed. Finally, when $D=\mathbb{T}^2$, the solution $u$ is recovered as the limit of Galerkin approximations. The proof relies on global existence of smooth solutions and weak-strong uniqueness arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14100 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The 2D Euler equations are well-posed for generic initial data in $L^2$ Galeati, Lucio Analysis of PDEs In this note we show the existence of a residual set (in the sense of Baire) of divergence free initial data $u_0\in L^2(D)$, $D=\mathbb{R}^2$ or $\mathbb{T}^2$, for which global existence and uniqueness of weak solutions to the incompressible 2D Euler equations holds. The associated solutions $u$ satisfy the energy balance and are recovered in the vanishing viscosity limit from solutions to 2D Navier-Stokes, which as a consequence cannot display anomalous dissipation of energy. Additionally, there exists a unique regular Lagrangian flow associated to such $u$, and the associated transport equation is well-posed. Finally, when $D=\mathbb{T}^2$, the solution $u$ is recovered as the limit of Galerkin approximations. The proof relies on global existence of smooth solutions and weak-strong uniqueness arguments. |
| title | The 2D Euler equations are well-posed for generic initial data in $L^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.14100 |