Regularity of solutions of the fractional porous medium flow

Abstract

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. More precisely, $$u_t=\nabla \cdot (u\nabla (-\Delta {)}^{-s}u),0<s<1.$$ The problem is posed in $\{x\in\ren, t\in \re\}$ 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. Here we establish the boundedness and $C^{\alpha }$ regularity of such weak solutions

55 pages, 2 figures

References in corpus (1)

• Regularity theory for nonlinear integral operators

Cited by in corpus (2)

• Well-posedness of a porous medium flow with fractional pressure in Sobolev spaces
• Global well-posedness for the homogeneous Landau equation