Existence of Weak Solutions for a General Porous Medium Equation with Nonlocal Pressure
Abstract
We study the general nonlinear diffusion equation {u_t=\nabla\cdot (u^{m1}\nabla (∆)^{s}u)} 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 nonnegative and that the problem is posed in the whole space. In this paper we prove the existence of weak solutions for all integrable initial data {u_0 ≥ 0} and for all exponents {m > 1} by developing a new approximation method that allows one to treat the range {m≥q 3}, which could not be covered by previous works. We also extend the class of initial data to include any nonnegative measure {μ} with finite mass. In passing from bounded initial data to measure data we make strong use of an L ^{1}{L^∞} smoothing effect and other functional estimates. Finite speed of propagation is established for all {m ≥q 2}, and this property implies the existence of free boundaries. The authors had already proved that finite propagation does not hold for {m < 2}.
 Publication:

Archive for Rational Mechanics and Analysis
 Pub Date:
 July 2019
 DOI:
 10.1007/s00205019013610
 Bibcode:
 2019ArRMA.233..451S