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

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

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

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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
\Bibitem{CafVaz15} \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 \mathnet{http://mi.mathnet.ru/aa1438} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3570960} \elib{http://elibrary.ru/item.asp?id=24849893} \transl \jour St. Petersburg Math. J. \yr 2016 \vol 27 \issue 3 \pages 437--460 \crossref{https://doi.org/10.1090/spmj/1397} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000373930300007} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84963533307} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. J. L. Vazquez, “The Dirichlet problem for the fractional $p$-Laplacian evolution equation”, J. Differ. Equ., 260:7 (2016), 6038–6056
3. J. Villa-Morales, “Instantaneous blow-up of semilinear non-autonomous equations with fractional diffusion”, Electron. J. Differ. Equ., 2017, 116
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
5. Y. Cheng, Sh. Gao, Yu. Wu, “Exact controllability of fractional order evolution equations in Banach spaces”, Adv. Differ. Equ., 2018, 332
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
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
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