August  2014, 7(4): 857-885. doi: 10.3934/dcdss.2014.7.857

Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators

1. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid

Received  November 2013 Published  February 2014

We report on recent progress in the study of nonlinear diffusion equations involving nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous medium equation, $\partial_t u +(-\Delta)^{s}(u^m)=0$, and some of its generalizations. Contrary to usual porous medium flows, the fractional version has infinite speed of propagation for all exponents $0 < s < 1$ and $m > 0$; on the other hand, it also generates an $L^1$-contraction semigroup which depends continuously on the exponent of fractional differentiation and the exponent of the nonlinearity.
    After establishing the general existence and uniqueness theory, the main properties are described: positivity, regularity, continuous dependence, a priori estimates, Schwarz symmetrization, among others. Self-similar solutions are constructed (fractional Barenblatt solutions) and they explain the asymptotic behaviour of a large class of solutions. In the fast diffusion range we study extinction in finite time and we find suitable special solutions. We discuss KPP type propagation. We also examine some related equations that extend the model and briefly comment on current work.
Citation: Juan-Luis Vázquez. Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 857-885. doi: 10.3934/dcdss.2014.7.857
References:
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References:
[1]

N. Alibaud, S. Cifani and E. Jakobsen, Continuous dependence estimates for nonlinear fractional convection-diffusion equations,, SIAM J. Math. Anal., 44 (2012), 603. doi: 10.1137/110834342. Google Scholar

[2]

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[3]

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[4]

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[5]

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[6]

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[7]

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[8]

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[10]

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[11]

I. Athanasopoulos and L. A. Caffarelli, Continuity of the temperature in boundary heat control problem,, Advances in Mathematics, 224 (2010), 293. doi: 10.1016/j.aim.2009.11.010. Google Scholar

[12]

T. Aubin, Problemes isoprimtriques et espaces de Sobolev,, J. Diff. Geom., 11 (1976), 573. Google Scholar

[13]

C. Bandle, Isoperimetric Inequalities and Applications,, Monographs and Studies in Mathematics, (1980). Google Scholar

[14]

G. I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium,, (in Russian) Prikl. Mat. Mekh., 16 (1952), 67. Google Scholar

[15]

G. I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics,, Cambridge Texts in Applied Mathematics, (1996). Google Scholar

[16]

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[17]

A. L. Bertozzi, J. L. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels,, Nonlinearity, 22 (2009), 683. doi: 10.1088/0951-7715/22/3/009. Google Scholar

[18]

A. Bertozzi, T. Laurent and F. Léger, Aggregation via Newtonian Potential and Aggregation Patches,, M3AS, 22 (2012). Google Scholar

[19]

P. Biler, C. Imbert and G. Karch, Barenblatt profiles for a nonlocal porous medium equation,, Comptes Rendus Mathematique, 349 (2011), 641. doi: 10.1016/j.crma.2011.06.003. Google Scholar

[20]

P. Biler, C. Imbert and G. Karch, Nonlocal porous medium equation: Barenblatt profiles and other weak solutions,, preprint , (2013). Google Scholar

[21]

P. Biler, G. Karch and R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions,, Comm. Math. Phys., 294 (2010), 145. doi: 10.1007/s00220-009-0855-8. Google Scholar

[22]

R. M. Blumenthal and R. K Getoor, Some theorems on stable processes,, Trans. Amer. Math. Soc., 95 (1960), 263. doi: 10.1090/S0002-9947-1960-0119247-6. Google Scholar

[23]

M. Bologna, P. Grigolini and C. Tsallis, Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions,, Physical Review E, 62 (2000). doi: 10.1103/PhysRevE.62.2213. Google Scholar

[24]

M. Bonforte and J. L. Vázquez, Global positivity estimates and Harnack inequalities for the fast diffusion equation,, J. Funct. Anal., 240 (2006), 399. doi: 10.1016/j.jfa.2006.07.009. Google Scholar

[25]

M. Bonforte and J. L. Vázquez, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations,, Advances in Math., 223 (2010), 529. doi: 10.1016/j.aim.2009.08.021. Google Scholar

[26]

M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities,, Proceedings Natl. Acad. Sci. USA, 107 (2010), 16459. doi: 10.1073/pnas.1003972107. Google Scholar

[27]

M. Bonforte and J. L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations,, Advances in Math., 250 (2014), 242. doi: 10.1016/j.aim.2013.09.018. Google Scholar

[28]

M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains,, preprint , (). Google Scholar

[29]

M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains. Part II,, in preparation., (). Google Scholar

[30]

J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s\to 1$ and applications,, J. Anal. Math., 87 (2002), 77. Google Scholar

[31]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. in Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[32]

X. Cabré and J. M. Roquejoffre, Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire,, C. R. Math. Acad. Sci. Paris, 347 (2009), 1361. doi: 10.1016/j.crma.2009.10.012. Google Scholar

[33]

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