Incompressible flow in porous media with fractional diffusion

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