Incompressible flow in porous media with fractional diffusion

TL;DR

This paper investigates heat transfer in incompressible fluids within porous media using fractional diffusion, establishing existence, uniqueness, decay, and attractor properties of solutions across various cases.

Contribution

It introduces new mathematical results on the existence, uniqueness, and long-term behavior of solutions for fractional diffusion models in porous media.

Findings

Formation of singularities with infinite energy.

Existence and uniqueness of strong solutions in sub-critical and critical cases.

Global existence of weak solutions and decay properties.

Abstract

In this paper we study the heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. We show formation of singularities with infinite energy and for finite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in $L^p$, for any $p\geq2$, and the asymptotic behavior is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critical dissipative case with $\alpha\in (1,2]$, we obtain the existence of the global attractor for the solutions in the space $H^s$ for any $s > (N/2)+1-\alpha$.