3,031 research outputs found
A fractional porous medium equation
We develop a theory of existence, uniqueness and regularity for a porous
medium equation with fractional diffusion, in , with ,
and . An -contraction semigroup is
constructed and the continuous dependence on data and exponent is established.
Nonnegative solutions are proved to be continuous and strictly positive for all
,
Porous medium equation with nonlocal pressure
We provide a rather complete description of the results obtained so far on
the nonlinear diffusion equation , which describes a flow through a porous medium driven by a
nonlocal pressure. We consider constant parameters and , we assume
that the solutions are non-negative, and the problem is posed in the whole
space. We present a theory of existence of solutions, results on uniqueness,
and relation to other models. As new results of this paper, we prove the
existence of self-similar solutions in the range when and , and the
asymptotic behavior of solutions when . The cases and were
rather well known.Comment: 24 pages, 2 figure
Stochastic models associated to a Nonlocal Porous Medium Equation
The nonlocal porous medium equation considered in this paper is a degenerate
nonlinear evolution equation involving a space pseudo-differential operator of
fractional order. This space-fractional equation admits an explicit,
nonnegative, compactly supported weak solution representing a probability
density function. In this paper we analyze the link between isotropic transport
processes, or random flights, and the nonlocal porous medium equation. In
particular, we focus our attention on the interpretation of the weak solution
of the nonlinear diffusion equation by means of random flights.Comment: Published at https://doi.org/10.15559/18-VMSTA112 in the Modern
Stochastics: Theory and Applications (https://vmsta.org/) by VTeX
(http://www.vtex.lt/
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