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


Authors: L. Caffarelli and J. L. Vázquez
Original publication: Algebra i Analiz, tom 27 (2015), nomer 3.
Journal: St. Petersburg Math. J. 27 (2016), 437-460
MSC (2010): Primary 35K55, 35K65, 35R11, 76S05
DOI: https://doi.org/10.1090/spmj/1397
Published electronically: March 30, 2016
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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.


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Additional Information

L. Caffarelli
Affiliation: School of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, Texas 78712-1082, Second affiliation: Institute for Computational Engineering and Sciences
Email: caffarel@math.utexas.edu

J. L. Vázquez
Affiliation: Universidad Autónoma de Madrid, Departamento de Matemáticas, 28049 Madrid, Spain
Email: juanluis.vazquez@uam.es

DOI: https://doi.org/10.1090/spmj/1397
Keywords: Porous medium equation, fractional Laplacian, nonlocal diffusion operator, H\"older regularity
Received by editor(s): January 6, 2015
Published electronically: March 30, 2016
Additional Notes: L. Caffarelli was funded by NSF Grant DMS-0654267 (Analytical and Geometrical Problems in Non Linear Partial Differential Equations), and J. L. Vázquez by Spanish Grant MTM2011-24696. Part of the work was done while both authors were visitors at the Isaac Newton Institute, Cambridge, during the Free Boundary programme 2014. We thank the institute for support and hospitality
Dedicated: Dedicated to N. N. Ural’tseva on her jubilee
Article copyright: © Copyright 2016 American Mathematical Society

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