Regularity of solutions of the fractional porous medium flow with exponent \(1/2\) (Q2797731)

The authors study regularity of solutions of the porous medium equation with nonlocal diffusion NEWLINENEWLINE\[NEWLINEu_t=\nabla\cdot(u\nabla(-\Delta)^{-1/2}u).NEWLINE\]NEWLINE NEWLINEA more general equation with \(s\in(0,1)\) instead of \(1/2\) has been thoroughly studied in previous papers. In particular, the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and the finite propagation property have been established but the Hölder regularity of solutions has not been proved in the case \(s=1/2\). To obtain that regularity, the authors combine De Giorgi type estimates with iterated geometric corrections that are needed to avoid the divergence of some energy integrals due to fractional long-range effects.