The $N$-dimensional gravity driven Muskat problem
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912894454595584 |
|---|---|
| author | Matioc, Bogdan-Vasile Prokert, Georg |
| author_facet | Matioc, Bogdan-Vasile Prokert, Georg |
| contents | We study the Muskat problem, which describes the motion of two immiscible, incompressible fluids in a homogeneous porous medium occupying the full space ${\mathbb{R}^{N+1}}$, $N \geq 2$, driven by gravity. The interface between the fluids is given as graph of a function over $\mathbb{R}^N$. The problem is reformulated as a nonlinear, nonlocal evolution problem for this function, involving singular integrals arising from potential representations of the velocity and pressure fields. Using results from harmonic analysis, we demonstrate that the evolution is of parabolic type in the open set identified by the Rayleigh-Taylor condition. We use the abstract theory of such problems to establish that the Muskat problem defines a semiflow on this set in all subcritical Sobolev spaces $H^s(\mathbb{R}^N)$, $s>s_c$, where ${s_c=1+N/2}$ is the critical exponent. We additionally obtain parabolic smoothing up to ${\rm C}^\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09786 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $N$-dimensional gravity driven Muskat problem Matioc, Bogdan-Vasile Prokert, Georg Analysis of PDEs 35R37, 35K55, 35Q35, 42B20 We study the Muskat problem, which describes the motion of two immiscible, incompressible fluids in a homogeneous porous medium occupying the full space ${\mathbb{R}^{N+1}}$, $N \geq 2$, driven by gravity. The interface between the fluids is given as graph of a function over $\mathbb{R}^N$. The problem is reformulated as a nonlinear, nonlocal evolution problem for this function, involving singular integrals arising from potential representations of the velocity and pressure fields. Using results from harmonic analysis, we demonstrate that the evolution is of parabolic type in the open set identified by the Rayleigh-Taylor condition. We use the abstract theory of such problems to establish that the Muskat problem defines a semiflow on this set in all subcritical Sobolev spaces $H^s(\mathbb{R}^N)$, $s>s_c$, where ${s_c=1+N/2}$ is the critical exponent. We additionally obtain parabolic smoothing up to ${\rm C}^\infty$. |
| title | The $N$-dimensional gravity driven Muskat problem |
| topic | Analysis of PDEs 35R37, 35K55, 35Q35, 42B20 |
| url | https://arxiv.org/abs/2602.09786 |