Well-posedness for Ohkitani model and long-time existence for surface quasi-geostrophic equations
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| Format: | Preprint |
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2023
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| _version_ | 1866929716212006912 |
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| author | Chae, Dongho Jeong, In-Jee Na, Jungkyoung Oh, Sung-Jin |
| author_facet | Chae, Dongho Jeong, In-Jee Na, Jungkyoung Oh, Sung-Jin |
| contents | We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani, $$\partial_t θ- \nabla^\perp \log(10+(-Δ)^{\frac12})θ\cdot \nabla θ= 0 ,$$ and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae, Constantin, Córdoba, Gancedo, and Wu in \cite{CCCGW}.
This well-posedness result can be applied to describe the long-time dynamics of the $δ$-SQG equations, defined by $$\partial_t θ+ \nabla^\perp (10+(-Δ)^{\frac12})^{-δ}θ\cdot \nabla θ= 0,$$ for all sufficiently small $δ>0$ depending on the size of the initial data. For the same range of $δ$, we establish global well-posedness of smooth solutions to the logarithmically dissipative counterpart: $$\partial_t θ+ \nabla^\perp (10+(-Δ)^{\frac12})^{-δ}θ\cdot \nabla θ+ \log(10+(-Δ)^{\frac12})θ= 0.$$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_02107 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Well-posedness for Ohkitani model and long-time existence for surface quasi-geostrophic equations Chae, Dongho Jeong, In-Jee Na, Jungkyoung Oh, Sung-Jin Analysis of PDEs Mathematical Physics We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani, $$\partial_t θ- \nabla^\perp \log(10+(-Δ)^{\frac12})θ\cdot \nabla θ= 0 ,$$ and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae, Constantin, Córdoba, Gancedo, and Wu in \cite{CCCGW}. This well-posedness result can be applied to describe the long-time dynamics of the $δ$-SQG equations, defined by $$\partial_t θ+ \nabla^\perp (10+(-Δ)^{\frac12})^{-δ}θ\cdot \nabla θ= 0,$$ for all sufficiently small $δ>0$ depending on the size of the initial data. For the same range of $δ$, we establish global well-posedness of smooth solutions to the logarithmically dissipative counterpart: $$\partial_t θ+ \nabla^\perp (10+(-Δ)^{\frac12})^{-δ}θ\cdot \nabla θ+ \log(10+(-Δ)^{\frac12})θ= 0.$$ |
| title | Well-posedness for Ohkitani model and long-time existence for surface quasi-geostrophic equations |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2308.02107 |