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

We study the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla\cdot(u\nabla (-Δ)^{-1/2}u).$ For definiteness, the problem is posed in $\{x\in\mathbb{R}^N, t\in \mathbb{R}\}$ with nonnegative initial data $u(x,0)$ that 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 $L^1$ data, for the more general family of equations $u_t=\nabla\cdot(u\nabla (-Δ)^{-s}u)$, $0<s<1$. Here we establish the $C^α$ regularity of such weak solutions 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 $[5]$. 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.