Nonlinear porous medium flow with fractional potential pressure

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Abstract

We study a porous medium equation, with nonlocal diffusion effects given by an inverse fractional Laplacian operator. We pose the problem in n-dimensional space for all t>0 with bounded and compactly supported initial data, and prove existence of a weak and bounded solution that propagates with finite speed, a property that is nor shared by other fractional diffusion models.

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An extension problem related to the fractional Laplacian

Luis A. Caffarelli, Luís Silvestre (2006)

The operator square root of the Laplacian \((-\lap)^{1/2}\) can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.

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Holder estimates for solutions of integro differential equations like the fractional laplace

Luís Silvestre (2006)

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Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian

Luís Silvestre, Luis A. Caffarelli, Sandro Salsa (2007)

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.