On well/ill-posedness for the generalized surface quasi-geostrophic equations in Hölder spaces
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| Format: | Preprint |
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2024
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| author | Choi, Young-Pil Jung, Jinwook Kim, Junha |
| author_facet | Choi, Young-Pil Jung, Jinwook Kim, Junha |
| contents | We establish the well/ill-posedness theories for the inviscid $α$-surface quasi-geostrophic ($α$-SQG) equations in Hölder spaces, where $α= 0$ and $α= 1$ correspond to the two-dimensional Euler equation in the vorticity formulation and SQG equation of geophysical significance, respectively. We first prove the local-in-time well-posedness of $α$-SQG equations in $C([0,T);C^{0,β}(\mathbb{R}^2))$ with $β\in (α,1)$ for some $T>0$. We then analyze the strong ill-posedness in $C^{0,α}(\mathbb{R}^2)$ constructing smooth solutions to the $α$-SQG equations that exhibit $C^{0,α}$--norm growth in a short time. In particular, we develop the nonexistence theory for $α$-SQG equations in $C^{0,α}(\mathbb{R}^2)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_01245 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On well/ill-posedness for the generalized surface quasi-geostrophic equations in Hölder spaces Choi, Young-Pil Jung, Jinwook Kim, Junha Analysis of PDEs We establish the well/ill-posedness theories for the inviscid $α$-surface quasi-geostrophic ($α$-SQG) equations in Hölder spaces, where $α= 0$ and $α= 1$ correspond to the two-dimensional Euler equation in the vorticity formulation and SQG equation of geophysical significance, respectively. We first prove the local-in-time well-posedness of $α$-SQG equations in $C([0,T);C^{0,β}(\mathbb{R}^2))$ with $β\in (α,1)$ for some $T>0$. We then analyze the strong ill-posedness in $C^{0,α}(\mathbb{R}^2)$ constructing smooth solutions to the $α$-SQG equations that exhibit $C^{0,α}$--norm growth in a short time. In particular, we develop the nonexistence theory for $α$-SQG equations in $C^{0,α}(\mathbb{R}^2)$. |
| title | On well/ill-posedness for the generalized surface quasi-geostrophic equations in Hölder spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.01245 |