Mild ill-posedness in $W^{1,\infty}$ for the incompressible porous media equation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913796737466368 |
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| author | Xie, Yaowei Yu, Huan |
| author_facet | Xie, Yaowei Yu, Huan |
| contents | In this paper, we establish the mild ill-posedness of 2D IPM equation in the critical Sobolev space $W^{1,\infty}$ when the initial data are small perturbations of stable profile $g(x_2).$ Consequently, instability can be inferred. Notably, our results are valid for arbitrary vertically stratified density profiles $g(x_2)$ without imposing any restrictions on the sign of $g'(x_2).$ From a physical perspective, since gravity acts downward, density profiles satisfying $g'(x_2) < 0$ typically correspond to stable configurations, whereas those with $g '(x_2) > 0$ are generally expected to be unstable. Surprisingly, our analysis uncovers an unexpected instability even when $g'(x_2) < 0$ and $g'(x_2)\in W^{2,\infty}(\mathbb{R})$. To the best of our knowledge, this work provides the first rigorous demonstration of IPM instability for vertically nonlinear density profiles, marking a significant departure from conventional physical expectations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23727 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mild ill-posedness in $W^{1,\infty}$ for the incompressible porous media equation Xie, Yaowei Yu, Huan Analysis of PDEs In this paper, we establish the mild ill-posedness of 2D IPM equation in the critical Sobolev space $W^{1,\infty}$ when the initial data are small perturbations of stable profile $g(x_2).$ Consequently, instability can be inferred. Notably, our results are valid for arbitrary vertically stratified density profiles $g(x_2)$ without imposing any restrictions on the sign of $g'(x_2).$ From a physical perspective, since gravity acts downward, density profiles satisfying $g'(x_2) < 0$ typically correspond to stable configurations, whereas those with $g '(x_2) > 0$ are generally expected to be unstable. Surprisingly, our analysis uncovers an unexpected instability even when $g'(x_2) < 0$ and $g'(x_2)\in W^{2,\infty}(\mathbb{R})$. To the best of our knowledge, this work provides the first rigorous demonstration of IPM instability for vertically nonlinear density profiles, marking a significant departure from conventional physical expectations. |
| title | Mild ill-posedness in $W^{1,\infty}$ for the incompressible porous media equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.23727 |