Abstract:We study the dynamics of compressible fluids in rotating heterogeneous porous media. The fluid flow is of {F}orchheimer-type and is subject to a mixed mass and volumetric flux boundary condition. The governing equations are reduced to a nonlinear partial differential equation for the pseudo-pressure. This parabolic-typed equation can be degenerate and/or singular in the spatial variables, the unknown and its gradient. We establish the $L^\alpha$-estimate for the solutions, for any positive number $\alpha$, in terms of the initial and boundary data and the angular speed of rotation. It requires new elliptic and parabolic Sobolev inequalities and trace theorem with multiple weights that are suitable to the nonlinear structure of the equation. The $L^\infty$-estimate is then obtained without imposing any conditions on the $L^\infty$-norms of the weights and the initial and boundary data.
| Comments: | 47 pages, improved presentation. to appear in Mathematical Methods in the Applied Sciences |
| Subjects: | Analysis of PDEs (math.AP) |
| MSC classes: | 35Q35, 76S05, 35B45, 35K20, 35K65, 35K67 |
| Cite as: | arXiv:2512.13959 [math.AP] |
| (or arXiv:2512.13959v2 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2512.13959 arXiv-issued DOI via DataCite |
|
| Related DOI: | https://doi.org/10.1002/mma.70807
DOI(s) linking to related resources |
From: Luan Hoang [view email]
[v1]
Mon, 15 Dec 2025 23:34:00 UTC (192 KB)
[v2]
Tue, 26 May 2026 01:36:54 UTC (195 KB)
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