Global well‐posedness for incompressible flow in porous media with partial diffusion or fractional diffusion

This paper examines the equation of heat transfer with fractional diffusion or partial diffusion of an incompressible fluid in a porous medium. We establish two main results. The first result is the global regularity for the equation with partial diffusion when the norm of initial datum inH 1 ( R 3 )is small. The second result is to show that for the supercritical case, the equation with fractional diffusion has a unique global solution provided that the norm of initial datum in the Besov spaceB ̇ p , ∞ s ( R 3 )with s ≥ 1 − α + 3 pis small.