Local Existence for the 2D Euler Equations in a Critical Sobolev Space
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912049778393088 |
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| author | Cozzi, Elaine Harrison, Nicholas |
| author_facet | Cozzi, Elaine Harrison, Nicholas |
| contents | In their seminal work, Bourgain and Li establish strong ill-posedness of the 2D incompressible Euler equations with vorticity in the critical Sobolev space $W^{s,p}(\mathbb{R}^2)$ for $sp=2$ and $p\in(1,\infty)$. In this note, we establish short-time existence of solutions with vorticity in the critical space $W^{2,1}(\mathbb{R}^2)$. Under the additional assumption that the initial vorticity is Dini continuous, we prove that the $W^{2,1}$-regularity of vorticity persists for all time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19418 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local Existence for the 2D Euler Equations in a Critical Sobolev Space Cozzi, Elaine Harrison, Nicholas Analysis of PDEs In their seminal work, Bourgain and Li establish strong ill-posedness of the 2D incompressible Euler equations with vorticity in the critical Sobolev space $W^{s,p}(\mathbb{R}^2)$ for $sp=2$ and $p\in(1,\infty)$. In this note, we establish short-time existence of solutions with vorticity in the critical space $W^{2,1}(\mathbb{R}^2)$. Under the additional assumption that the initial vorticity is Dini continuous, we prove that the $W^{2,1}$-regularity of vorticity persists for all time. |
| title | Local Existence for the 2D Euler Equations in a Critical Sobolev Space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.19418 |