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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.01682 |
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| _version_ | 1866916869877792768 |
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| author | Na, Jungkyoung |
| author_facet | Na, Jungkyoung |
| contents | In this paper, we establish the existence of global self-similar solutions to the 3D Muskat equation when the two fluids have the same viscosity but different densities. These self-similar solutions are globally defined in both space and time, with exact cones as their initial data. Furthermore we estimate the difference between our self-similar solutions and solutions of the linearized equation around the flat interface in terms of critical spaces and some weighted $\dot{W}^{k,\infty}(\mathbb{R}^2)$ spaces for $k=1,2$. The main ingredients of the proof are new estimates in the sense of $\dot{H}^{s_1}(\mathbb{R}^2) \cap \dot{H}^{s_2}(\mathbb{R}^2)$ with $3/2<s_1<2<s_2<3$, which is continuously embedded in critical spaces for the 3D Muskat problem: $\dot{H}^2(\mathbb{R}^2)$ and $\dot{W}^{1,\infty}(\mathbb{R}^2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01682 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global self-similar solutions for the 3D Muskat equation Na, Jungkyoung Analysis of PDEs In this paper, we establish the existence of global self-similar solutions to the 3D Muskat equation when the two fluids have the same viscosity but different densities. These self-similar solutions are globally defined in both space and time, with exact cones as their initial data. Furthermore we estimate the difference between our self-similar solutions and solutions of the linearized equation around the flat interface in terms of critical spaces and some weighted $\dot{W}^{k,\infty}(\mathbb{R}^2)$ spaces for $k=1,2$. The main ingredients of the proof are new estimates in the sense of $\dot{H}^{s_1}(\mathbb{R}^2) \cap \dot{H}^{s_2}(\mathbb{R}^2)$ with $3/2<s_1<2<s_2<3$, which is continuously embedded in critical spaces for the 3D Muskat problem: $\dot{H}^2(\mathbb{R}^2)$ and $\dot{W}^{1,\infty}(\mathbb{R}^2)$. |
| title | Global self-similar solutions for the 3D Muskat equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.01682 |