Incompressible flow in porous media with fractional diffusion

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 $p\ge 2$, 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 $s>(N/2)+1-\alpha$.

References in corpus (1)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

Cited by in corpus (9)

• On the loss of continuity for super-critical drift-diffusion equations
• Unique Ergodicity for Fractionally Dissipated, Stochastically Forced 2D Euler Equations
• On Moffatt's magnetic relaxation equations
• Non-uniqueness in law for Boussinesq system forced by random noise
• A Remark on the Global Well-posedness of a Modified Critical Quasi-geostrophic Equation
• Spatial-decay of solutions to the quasi-geostrophic equation with the critical and the super-critical dissipation
• On the regularity issues of a class of drift-diffusion equations with nonlocal diffusion
• Entropy solutions to macroscopic IPM
• A local-in-time theory for singular SDEs with applications to fluid models with transport noise