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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2410.01297 |
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| _version_ | 1866913526926278656 |
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| author | Bianchini, Roberta Córdoba, Diego Martínez-Zoroa, Luis |
| author_facet | Bianchini, Roberta Córdoba, Diego Martínez-Zoroa, Luis |
| contents | We prove the non-existence and strong ill-posedness of the Incompressible Porous Media (IPM) equation for initial data that are small $H^2(\mathbb{R}^2)$ perturbations of the linearly stable profile $-x_2$. A remarkable novelty of the proof is the construction of an $H^2$ perturbation, which solves the IPM equation and neutralizes the stabilizing effect of the background profile near the origin, where a strong deformation leading to non-existence in $H^2$ is created. This strong deformation is achieved through an iterative procedure inspired by the work of Córdoba and Mart\'ınez-Zoroa (Adv. Math. 2022). However, several differences - beyond purely technical aspects - arise due to the anisotropic and, more importantly, to the partially dissipative nature of the equation, adding further challenges to the analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01297 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non Existence and Strong Ill-Posedness in $H^2$ for the Stable IPM Equation Bianchini, Roberta Córdoba, Diego Martínez-Zoroa, Luis Analysis of PDEs We prove the non-existence and strong ill-posedness of the Incompressible Porous Media (IPM) equation for initial data that are small $H^2(\mathbb{R}^2)$ perturbations of the linearly stable profile $-x_2$. A remarkable novelty of the proof is the construction of an $H^2$ perturbation, which solves the IPM equation and neutralizes the stabilizing effect of the background profile near the origin, where a strong deformation leading to non-existence in $H^2$ is created. This strong deformation is achieved through an iterative procedure inspired by the work of Córdoba and Mart\'ınez-Zoroa (Adv. Math. 2022). However, several differences - beyond purely technical aspects - arise due to the anisotropic and, more importantly, to the partially dissipative nature of the equation, adding further challenges to the analysis. |
| title | Non Existence and Strong Ill-Posedness in $H^2$ for the Stable IPM Equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.01297 |