Regularity of solutions of the fractional porous medium flow
Summary: We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is \[ u_t=\nabla\cdot(u\nabla (-\Delta)^{-s}u), \quad \;0<s<1. \] 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. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and \(C^{\alpha}\) regularity of such weak solutions. Finally, we extend the existence theory to all nonnegative and integrable initial data.
scientific article; zbMATH DE number 6203605
scientific article