# Regularity of solutions of the fractional porous medium flow

Luis Caffarelli; Fernando Soria; Juan Luis Vázquez

Journal of the European Mathematical Society (2013)

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

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topCaffarelli, 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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