The Derivative Structure for a Quadratic Nonlinearity and Uniqueness for SQG
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912017235836928 |
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| author | Iwabuchi, Tsukasa |
| author_facet | Iwabuchi, Tsukasa |
| contents | We study the two-dimensional surface quasi-geostrophic equation on a bounded domain with a smooth boundary. Motivated by the three-dimensional incompressible Navier-Stokes equations and previous results in the entire space $\mathbb R^2$, we demonstrate that the uniqueness of the mild solution holds in $L^2$. For the proof, we provide a method for handling fractional Laplacians in nonlinear problems, and develop an approach to derive second-order derivativesfor the nonlinear term involving fractional derivatives of the Dirichlet Laplacian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03955 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Derivative Structure for a Quadratic Nonlinearity and Uniqueness for SQG Iwabuchi, Tsukasa Analysis of PDEs 35Q35, 35Q86 We study the two-dimensional surface quasi-geostrophic equation on a bounded domain with a smooth boundary. Motivated by the three-dimensional incompressible Navier-Stokes equations and previous results in the entire space $\mathbb R^2$, we demonstrate that the uniqueness of the mild solution holds in $L^2$. For the proof, we provide a method for handling fractional Laplacians in nonlinear problems, and develop an approach to derive second-order derivativesfor the nonlinear term involving fractional derivatives of the Dirichlet Laplacian. |
| title | The Derivative Structure for a Quadratic Nonlinearity and Uniqueness for SQG |
| topic | Analysis of PDEs 35Q35, 35Q86 |
| url | https://arxiv.org/abs/2409.03955 |