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

Abstract:

The object of study is the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla \cdot (u\nabla (-\Delta )^{-1/2}u).$ For definiteness, the problem is posed in $\{x\in \mathbb {R}^N, t\in \mathbb {R}\}$ with nonnegative initial data $u(x,0)$ that is integrable and decays 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 $L^1$ data, for the more general family of equations $u_t=\nabla \cdot (u\nabla (-\Delta )^{-s}u)$, $0<s<1$.

Here, the $C^\alpha$ regularity of such weak solutions is established in the difficult fractional exponent case $s=1/2$. For the other fractional exponents $s\in (0,1)$ this Hölder regularity has been proved in an earlier paper. Continuity was under question because the nonlinear differential operator has first-order differentiation. 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.