Recent Trends in Nonlinear Diffusion

# Some current problems involving fractional Laplacians. Review

abstract: We review some topics in nonlinear diffusion involving nonlocal (long-range) effects. We describe two models of flow in porous media including nonlocal (long-range) diffusion effects. The first model is based on Darcy's law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density. We use entropy methods to show that they describe the asymptotic behaviour of a wide class of solutions. The second model is more in the spirit of fractional Laplacian flows, but nonlinear. Contrary to usual Porous Medium flows (PME in the sequel), it has infinite speed of propagation. Similarly to them, an $$L^1$$-contraction semigroup is constructed and it depends continuously on the exponent of fractional derivation and the exponent of the nonlinearity. The existence of new Barenblatt solutions is described.