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

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.