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Algebra i Analiz, 2015, Volume 27, Issue 3, Pages 125–156 (Mi aa1438)  

This article is cited in 7 scientific papers (total in 7 papers)

Research Papers

Regularity of solutions of the fractional porous medium flow with exponent $1/2$

L. Caffarelliab, J. L. Vázquezc

a Institute for Computational Engineering and Sciences, USA
b School of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, Texas 78712-1082, USA
c Universidad Autónoma de Madrid, Departamento de Matemáticas, 28049, Madrid, Spain

Abstract: The object of study is the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla\cdot(u\nabla(-\Delta)^{-1/2}u)$. For definiteness, the problem is posed in $\{x\in\mathbb R^N, t\in\mathbb R\}$ with nonnegative initial data $u(x,0)$ that is integrable and decays at infinity. Previous papers have established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation, as well as the boundedness of nonnegative solutions with $L^1$ data, for the more general family of equations $u_t=\nabla\cdot(u\nabla(-\Delta)^{-s}u)$, $0<s<1$.
Here, the $C^\alpha$ regularity of such weak solutions is established in the difficult fractional exponent case $s=1/2$. For the other fractional exponents $s\in(0,1)$ this Hölder regularity has been proved in an earlier paper. Continuity was under question because the nonlinear differential operator has first-order differentiation. The method combines delicate De Giorgi type estimates with iterated geometric corrections that are needed to avoid the divergence of some essential energy integrals due to fractional long-range effects.

Keywords: porous medium equation, fractional Laplacian, nonlocal diffusion operator, Hölder regularity.

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English version:
St. Petersburg Mathematical Journal, 2016, 27:3, 437–460

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Received: 06.01.2015

Citation: L. Caffarelli, J. L. Vázquez, “Regularity of solutions of the fractional porous medium flow with exponent $1/2$”, Algebra i Analiz, 27:3 (2015), 125–156; St. Petersburg Math. J., 27:3 (2016), 437–460

Citation in format AMSBIB
\by L.~Caffarelli, J.~L.~V\'azquez
\paper Regularity of solutions of the fractional porous medium flow with exponent~$1/2$
\jour Algebra i Analiz
\yr 2015
\vol 27
\issue 3
\pages 125--156
\jour St. Petersburg Math. J.
\yr 2016
\vol 27
\issue 3
\pages 437--460

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    This publication is cited in the following articles:
    1. D. Stan, F. del Teso, J. L. Vazquez, “Finite and infinite speed of propagation for porous medium equations with nonlocal pressure”, J. Differ. Equ., 260:2 (2016), 1154–1199  crossref  mathscinet  zmath  isi  scopus
    2. J. L. Vazquez, “The Dirichlet problem for the fractional $p$-Laplacian evolution equation”, J. Differ. Equ., 260:7 (2016), 6038–6056  crossref  mathscinet  zmath  isi  scopus
    3. J. Villa-Morales, “Instantaneous blow-up of semilinear non-autonomous equations with fractional diffusion”, Electron. J. Differ. Equ., 2017, 116  mathscinet  zmath  isi
    4. S. Lisini, E. Mainini, A. Segatti, “A gradient flow approach to the porous medium equation with fractional pressure”, Arch. Ration. Mech. Anal., 227:2 (2018), 567–606  crossref  mathscinet  zmath  isi  scopus
    5. Y. Cheng, Sh. Gao, Yu. Wu, “Exact controllability of fractional order evolution equations in Banach spaces”, Adv. Differ. Equ., 2018, 332  crossref  mathscinet  isi  scopus
    6. Quoc-Hung Nguyen, J. L. Vazquez, “Porous medium equation with nonlocal pressure in a bounded domain”, Commun. Partial Differ. Equ., 43:10 (2018), 1502–1539  crossref  mathscinet  zmath  isi  scopus
    7. D. Stan, F. Del Teso, J. L. Vazquez, “Existence of Weak Solutions For a General Porous Medium Equation With Nonlocal Pressure”, Arch. Ration. Mech. Anal., 233:1 (2019), 451–496  crossref  zmath  isi  scopus
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