Finite time blow-up in a 1D model of the incompressible porous media equation
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916718973026304 |
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| author | Kiselev, Alexander Sarsam, Naji A. |
| author_facet | Kiselev, Alexander Sarsam, Naji A. |
| contents | We derive a PDE that models the behavior of a boundary layer solution to the incompressible porous media (IPM) equation posed on the 2D periodic half-plane. This 1D IPM model is a transport equation with a non-local velocity similar to the well-known Córdoba-Córdoba-Fontelos (CCF) equation. We discuss how this modification of the CCF equation can be regarded as a reasonable model for solutions to the IPM equation. Working in the class of bounded smooth periodic data, we then show local well-posedness for the 1D IPM model as well as finite time blow-up for a class of initial data. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_16376 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite time blow-up in a 1D model of the incompressible porous media equation Kiselev, Alexander Sarsam, Naji A. Analysis of PDEs 76B03, 35Q35, 76S05 We derive a PDE that models the behavior of a boundary layer solution to the incompressible porous media (IPM) equation posed on the 2D periodic half-plane. This 1D IPM model is a transport equation with a non-local velocity similar to the well-known Córdoba-Córdoba-Fontelos (CCF) equation. We discuss how this modification of the CCF equation can be regarded as a reasonable model for solutions to the IPM equation. Working in the class of bounded smooth periodic data, we then show local well-posedness for the 1D IPM model as well as finite time blow-up for a class of initial data. |
| title | Finite time blow-up in a 1D model of the incompressible porous media equation |
| topic | Analysis of PDEs 76B03, 35Q35, 76S05 |
| url | https://arxiv.org/abs/2412.16376 |