Continuity of the solution map of some active scalar equations in Hölder and Zygmund spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912115344801792 |
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| author | Magaña, Marc |
| author_facet | Magaña, Marc |
| contents | We prove that the solution map for a family of non-linear transport equations in $\mathbb{R}^n$, with a velocity field given by the convolution of the density with a kernel that is smooth away from the origin and homogeneous of degree $-(n-1)$, is continuous in both the little Hölder class and the little Zygmund class. For particular choices of the kernel, one recovers well-known equations such as the 2D Euler or the 3D quasi-geostrophic equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19057 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuity of the solution map of some active scalar equations in Hölder and Zygmund spaces Magaña, Marc Analysis of PDEs 35Q35 (Primary), 35F20, 35B30 (Secondary) We prove that the solution map for a family of non-linear transport equations in $\mathbb{R}^n$, with a velocity field given by the convolution of the density with a kernel that is smooth away from the origin and homogeneous of degree $-(n-1)$, is continuous in both the little Hölder class and the little Zygmund class. For particular choices of the kernel, one recovers well-known equations such as the 2D Euler or the 3D quasi-geostrophic equations. |
| title | Continuity of the solution map of some active scalar equations in Hölder and Zygmund spaces |
| topic | Analysis of PDEs 35Q35 (Primary), 35F20, 35B30 (Secondary) |
| url | https://arxiv.org/abs/2410.19057 |