Existence of Weak Solutions for a General Porous Medium Equation with Nonlocal Pressure
We study the general nonlinear diffusion equation ut = r· (u r(−Δ)−su) that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters m> 1 and 0 <s< 1, we assume that the solutions are non-negative and the problem is posed in the whole...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2019 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/710948 |
| Acceso en línea: | http://hdl.handle.net/10486/710948 https://dx.doi.org/10.1007/s00205-019-01361-0 |
| Access Level: | acceso abierto |
| Palabra clave: | Nonlinear fractional diffusion fractional Laplacian existence of weak solutions energy estimates speed of propagation smoothing effect numerical simulations Matemáticas |
| Sumario: | We study the general nonlinear diffusion equation ut = r· (u r(−Δ)−su) that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters m> 1 and 0 <s< 1, we assume that the solutions are non-negative and the problem is posed in the whole space. In this paper we prove existence of weak solutions for all integrable initial data u0 ≥ 0 and for all exponents m> 1 by developing a new approximation method that allows to treat the range m ≥ 3 that could not be covered by previous works. We also extend the class of initial data to include any non-negative measure μ with finite mass. In passing from bounded initial data to measure data we make strong use of an L1-L∞ smoothing effect and other functional estimates. Finite speed of propagation is established for all m ≥ 2, and this property implies the existence of free boundaries. The authors had already proved that finite propagation does not hold for m< 2 |
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