Regularity of solutions of the fractional porous medium flow

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 -