Regularity of solutions of the fractional porous medium flow with exponent $1/2$

Abstract

We study the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is ut = ∇·(u∇(−∆)−1/2u). For definiteness, the problem is posed in {x ∈ RN , t ∈ R} with nonnegative initial data u(x, 0) that are integrable and decay at infinity. Previous papers have established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation, as well as the boundedness of nonnegative solutions with L1 data, for the more general family of equations ut = ∇ · (u∇(−∆)−su), 0 < s < 1. Here we establish the Cα regularity of such weak solutions in the difficult fractional exponent case s = 1/2. For the other fractional exponents s ∈ (0, 1) this Holder regularity has been proved in [5]. The method combines delicate De Giorgi type estimates with iterated geometric corrections that are needed to avoid the divergence of some essential energy integrals due to fractional long-range effects. ∗University of Texas; caffarel@math.utexas.edu †Universidad Autonoma de Madrid; juanluis.vazquez@uam.es