# Regularity of solutions of the fractional porous medium flow

• Volume: 015, Issue: 5, page 1701-1746
• ISSN: 1435-9855

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## Abstract

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We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is ${u}_{t}=\nabla ·\left(u\nabla {\left(-\Delta \right)}^{-s}u\right),\phantom{\rule{1.0em}{0ex}}\phantom{\rule{4pt}{0ex}}0. The problem is posed in $\left\{x\in {ℝ}^{n},t\in ℝ\right\}$ with nonnegative initial data $u\left(x,0\right)$ 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.

## How to cite

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Caffarelli, Luis, Soria, Fernando, and Vázquez, Juan Luis. "Regularity of solutions of the fractional porous medium flow." Journal of the European Mathematical Society 015.5 (2013): 1701-1746. <http://eudml.org/doc/277300>.

@article{Caffarelli2013,
abstract = {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 $\lbrace x\in \mathbb \{R\}^n, t\in \mathbb \{R\}\rbrace$ 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.},
author = {Caffarelli, Luis, Soria, Fernando, Vázquez, Juan Luis},
journal = {Journal of the European Mathematical Society},
keywords = {porous medium equation; fractional Laplacian; nonlocal operator; regularity; porous medium equation; fractional Laplacian; nonlocal operator; regularity},
language = {eng},
number = {5},
pages = {1701-1746},
publisher = {European Mathematical Society Publishing House},
title = {Regularity of solutions of the fractional porous medium flow},
url = {http://eudml.org/doc/277300},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Caffarelli, Luis
AU - Soria, Fernando
AU - Vázquez, Juan Luis
TI - Regularity of solutions of the fractional porous medium flow
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 5
SP - 1701
EP - 1746
AB - 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 $\lbrace x\in \mathbb {R}^n, t\in \mathbb {R}\rbrace$ 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.
LA - eng
KW - porous medium equation; fractional Laplacian; nonlocal operator; regularity; porous medium equation; fractional Laplacian; nonlocal operator; regularity
UR - http://eudml.org/doc/277300
ER -

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