625 research outputs found
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
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
,
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