Regularity of solutions of the fractional porous medium flow
type: Item
instance of: scholarly article
title: Regularity of solutions of the fractional porous medium flow
zbMATH Open document ID: 1292.35312
DOI: 10.4171/JEMS/401
author: Luis Caffarelli
author: F. Soria
author: Juan Luis Vázquez
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2013-09-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1201.6048
review text: 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.
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 76S05
zbMATH DE Number: 6203605
zbMATH Keywords: porous medium equation
zbMATH Keywords: fractional Laplacian
zbMATH Keywords: nonlocal operator
zbMATH Keywords: regularity
MaRDI profile type: Publication
arXiv ID: 1201.6048
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description: scientific article; zbMATH DE number 6203605