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:
[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.

[2]

N. Alibaud, S. Cifani and E. Jakobsen, Optimal continuous dependence estimates for fractional degenerate parabolic equations,, , ().

[3]

L. Ambrosio, E. Mainini and S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices,, Annales IHP, 28 (2011), 217. doi: 10.1016/j.anihpc.2010.11.006.

[4]

L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity,, Comm. Pure Appl. Math., 61 (2008), 1495. doi: 10.1002/cpa.20223.

[5]

F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo, Nonlocal Diffusion Problems,, AMS Mathematical Surveys and Monographs, (2010).

[6]

D. Applebaum, Lévy Processes and Stochastic Calculus,, Second edition, (2009). doi: 10.1017/CBO9780511809781.

[7]

D. G. Aronson, The porous medium equation,, in Nonlinear Diffusion Problems (Montecatini Terme, 1224 (1985), 1. doi: 10.1007/BFb0072687.

[8]

D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation,, Trans. Amer. Math. Soc., 280 (1983), 351. doi: 10.1090/S0002-9947-1983-0712265-1.

[9]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5.

[10]

I. Athanasopoulos and L. A. Caffarelli, Optimal regularity of lower dimensional obstacle problems,, Zap. Nauchn. Se. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), 49. doi: 10.1007/s10958-005-0496-1.

[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.

[12]

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

[13]

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

[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.

[15]

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

[16]

J. Bertoin, Lévy Processes,, Cambridge Tracts in Mathematics, (1996).

[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.

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[33]

X. Cabré and J. M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations,, Comm. Math. Phys., 320 (2013), 679. doi: 10.1007/s00220-013-1682-5.

[34]

L. A. Caffarelli, Further regularity for the Signorini problem,, Comm. Partial Differential Equations, 4 (1979), 1067. doi: 10.1080/03605307908820119.

[35]

L. Caffarelli, C.-H. Chan and A. Vasseur, Regularity theory for nonlinear integral operators,, J. Amer. Math. Soc., 24 (2011), 849. doi: 10.1090/S0894-0347-2011-00698-X.

[36]

L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian,, Invent. Math., 171 (2008), 425. doi: 10.1007/s00222-007-0086-6.

[37]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306.

[38]

L. A. Caffarelli, F. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow,, J. Eur. Math. Soc. (JEMS), 15 (2013), 1701. doi: 10.4171/JEMS/401.

[39]

L. A. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure,, Arch. Rational Mech. Anal., 202 (2011), 537. doi: 10.1007/s00205-011-0420-4.

[40]

L. A. Caffarelli and J. L. Vázquez, Asymptotic behaviour of a porous medium equation with fractional diffusion,, Discrete Cont. Dyn. Systems-A, 29 (2011), 1393.

[41]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann. of Math., 171 (2010), 1903. doi: 10.4007/annals.2010.171.1903.

[42]

A. Capella, J. Davila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non local semilinear equations,, Comm. Partial Diff. Eq., 36 (2011), 1353. doi: 10.1080/03605302.2011.562954.

[43]

J. A. Carrillo, Y. Huang and J. L. Vazquez, in, preparation., ().

[44]

Z. Q. Chen, P. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian,, J. Eur. Math. Soc. (JEMS), 12 (2010), 1307. doi: 10.4171/JEMS/231.

[45]

S. Cifani and E. R. Jakobsen, Entropy solution theory for fractional degenerate convection-diffusion equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 413. doi: 10.1016/j.anihpc.2011.02.006.

[46]

S. Cifani, E. R. Jakobsen and K. H. Karlsen, The discontinuous Galerkin method for fractional degenerate convection-diffusion equations,, BIT, 51 (2011), 809. doi: 10.1007/s10543-011-0327-3.

[47]

J. S. Chapman, J. Rubinstein and M. Schatzman, A mean-field model for superconducting vortices,, Eur. J. Appl. Math., 7 (1996), 97. doi: 10.1017/S0956792500002242.

[48]

R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC, (2004).

[49]

E. B. Davies, Heat Kernels and Spectral Theory,, Cambridge Tracts in Mathematics, (1990).

[50]

A. de Pablo and J. L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation,, J. Differential Equations, 93 (1991), 19. doi: 10.1016/0022-0396(91)90021-Z.

[51]

A. De Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A fractional porous medium equation,, Advances in Mathematics, 226 (2011), 1378. doi: 10.1016/j.aim.2010.07.017.

[52]

A. De Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A general fractional porous medium equation,, Comm. Pure Appl. Math., 65 (2012), 1242. doi: 10.1002/cpa.21408.

[53]

A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, Classical solutions for a logarithmic fractional diffusion equation,, to appear in Journal de Math. Pures Appliquées, ().

[54]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces,, preprint, (2011).

[55]

G. Duvaut and J.-L. Lions, Les Inéquations en Mechanique et en Physique,, Travaux et Recherches Mathématiques, (1972).

[56]

W. E, Dynamics of vortex-liquids in Ginzburg-Landau theories with applications to superconductivity,, Phys. Rev. B, 50 (1994), 1126.

[57]

R. A. Fisher, The wave of advance of advantagenous genes,, Ann. Eugenics, 7 (1937), 355.

[58]

R. K. Getoor, First passage times for symmetric stable processes in space,, Trans. Amer. Math. Soc., 101 (1961), 75. doi: 10.1090/S0002-9947-1961-0137148-5.

[59]

A. K. Head, Dislocation group dynamics II. Similarity solutions of the continuum approximation,, Phil. Mag., 26 (1972), 65.

[60]

Y. H. Huang, Explicit barenblatt profiles for fractional porous medium equations,, preprint, (2013).

[61]

M. D. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps,, Comm. Pure Applied Math., 62 (2009), 198. doi: 10.1002/cpa.20253.

[62]

M. D. Jara, T. Komorowski and S. Olla, Limit theorems for additive functionals of a Markov chain,, Ann. Appl. Probab., 19 (2009), 2270. doi: 10.1214/09-AAP610.

[63]

M. Jara, Hydrodynamic limit Of particle systems with long jumps,, , ().

[64]

M. Jara, C. Landim and S. Sethuraman, Nonequilibrium fluctuations for a tagged particle in mean-zero one-dimensional zero-range processes,, Probab. Theory Relat. Fields, 145 (2009), 565. doi: 10.1007/s00440-008-0178-2.

[65]

M. Kassmann, A priori estimates for integro-differential operators with measurable kernels,, Calc. Var., 34 (2009), 1. doi: 10.1007/s00526-008-0173-6.

[66]

J. King and P. McCabe, On the Fisher-KPP equation with fast nonlinear diffusion,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 2529. doi: 10.1098/rspa.2003.1134.

[67]

A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation,, Invent. Math., 167 (2007), 445. doi: 10.1007/s00222-006-0020-3.

[68]

A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Etude de l'équation de diffusion avec accroissement de la quantité de matière, et son application à un problème biologique,, Bjul. Moskowskogo Gos. Univ., 17 (1937), 1.

[69]

N. S. Landkof, Foundations of Modern Potential Theory,, Translated from the Russian by A. P. Doohovskoy, (1972).

[70]

E. K. Lenzi, R. S. Mendes and C. Tsallis, Crossover in diffusion equation: Anomalous and normal behaviors,, Physical Review E, 67 (2003). doi: 10.1103/PhysRevE.67.031104.

[71]

F. H. Lin and P. Zhang, On the hydrodynamic limit of Ginzburg-Landau vortices,, Discrete Cont. Dyn. Systems, 6 (2000), 121. doi: 10.3934/dcds.2000.6.121.

[72]

V. I. Mazya and T. O. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces,, Journal Funct. Anal., 195 (2002), 230. doi: 10.1006/jfan.2002.3955.

[73]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations,, Arch. Ration. Mech. Anal., 199 (2011), 493. doi: 10.1007/s00205-010-0354-2.

[74]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach,, Physics Reports, 339 (2000), 1. doi: 10.1016/S0370-1573(00)00070-3.

[75]

R. H. Nochetto, E. Otarola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis,, , ().

[76]

S. Serfaty and J. L. Vazquez, A mean field equation as limit of nonlinear diffusion with fractional laplacian operators,, Calc. Var. PDEs, (2013). doi: 10.1007/s00526-013-0613-9.

[77]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators,, to appear in Proc. Roy. Soc. Edinburgh Sect. A. Available from: , (): 12.

[78]

A. Signorini, Questioni di elasticità non linearizzata e semilinearizzata,, Rendiconti di Matematica e delle sue Applicazioni, 18 (1959), 95.

[79]

L. E. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional Laplace,, Indiana Univ. Math. J., 55 (2006), 1155. doi: 10.1512/iumj.2006.55.2706.

[80]

L. E. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, Comm. Pure Appl. Math., 60 (2007), 6. doi: 10.1002/cpa.20153.

[81]

D. Stan and J. L. Vázquez, The Fisher-KPP equation with nonlinear fractional diffusion,, submitted, (2013).

[82]

D. Stan, F. del Teso and J. L. Vázquez, Finite and infinite speed of propagation for porous medium equations with fractional pressure,, Comptes Rendus Acad. Sci. Paris, 352 (2014), 123.

[83]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).

[84]

G. Talenti, Elliptic equations and rearrangements,, Ann. Scuola Norm. Sup. (4), 3 (1976), 697.

[85]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl. (4), 110 (1976), 353. doi: 10.1007/BF02418013.

[86]

F. del Teso, Finite difference method for a fractional porous medium equation,, to appear in Calcolo, (2013). doi: 10.1007/s10092-013-0103-7.

[87]

F. del Teso and J. L. Vázquez, Finite difference method for a general fractional porous medium equation,, , ().

[88]

E. Valdinoci, From the long jump random walk to the fractional Laplacian,, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33.

[89]

J. L. Vázquez, Symétrisation pour $u_t=\Delta\varphi(u)$ et applications,, C. R. Acad. Sc. Paris, 295 (1982), 71.

[90]

J. L. Vázquez, Smoothing And Decay Estimates For Nonlinear Diffusion Equations. Equations Of Porous Medium Type,, Oxford Lecture Series in Mathematics and its Applications, (2006). doi: 10.1093/acprof:oso/9780199202973.001.0001.

[91]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs, (2007).

[92]

J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators,, in Nonlinear partial differential equations: the Abel Symposium 2010 (ed. H. Kenneth), (2010), 271.

[93]

J. L. Vázquez, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type,, to appear in J. Europ. Math. Soc.; , (2013).

[94]

J. L. Vázquez, A. de Pablo, F. Quirós and A. Rodríguez, Classical solutions and higher regularity for nonlinear fractional diffusion equations;, , ().

[95]

J. L. Vázquez and B. Volzone, Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type,, to appear in J. Math. Pures Appl.; , ().

[96]

J. L. Vázquez and B. Volzone, Optimal estimates for Fractional Fast diffusion equations,, submitted, ().

[97]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous Diffusion: A tutorial,, in Order and Chaos, (2008).

[98]

H. Weinberger, Symmetrization in Uniformly Elliptic Problems,, in 1962 Studies in Mathematical Analysis and Related Topics, (1962), 424.

[99]

H. Weitzner and G. M. Zaslavsky, Some applications of fractional equations. Chaotic transport and complexity in classical and quantum dynamics,, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273. doi: 10.1016/S1007-5704(03)00049-2.

[100]

W. A. Woyczyński, Lévy processes in the physical sciences,, in Lévy Processes - Theory and Applications, (2001), 241.

[101]

Ya. B. Zel'dovich and A. S. Kompanyeets, Towards a theory of heat conduction with thermal conductivity depending on the temperature,, in Collection of Papers Dedicated to 70th Anniversary of A. F. Ioffe, (1950), 61.

show all references

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.

[2]

N. Alibaud, S. Cifani and E. Jakobsen, Optimal continuous dependence estimates for fractional degenerate parabolic equations,, , ().

[3]

L. Ambrosio, E. Mainini and S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices,, Annales IHP, 28 (2011), 217. doi: 10.1016/j.anihpc.2010.11.006.

[4]

L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity,, Comm. Pure Appl. Math., 61 (2008), 1495. doi: 10.1002/cpa.20223.

[5]

F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo, Nonlocal Diffusion Problems,, AMS Mathematical Surveys and Monographs, (2010).

[6]

D. Applebaum, Lévy Processes and Stochastic Calculus,, Second edition, (2009). doi: 10.1017/CBO9780511809781.

[7]

D. G. Aronson, The porous medium equation,, in Nonlinear Diffusion Problems (Montecatini Terme, 1224 (1985), 1. doi: 10.1007/BFb0072687.

[8]

D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation,, Trans. Amer. Math. Soc., 280 (1983), 351. doi: 10.1090/S0002-9947-1983-0712265-1.

[9]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5.

[10]

I. Athanasopoulos and L. A. Caffarelli, Optimal regularity of lower dimensional obstacle problems,, Zap. Nauchn. Se. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), 49. doi: 10.1007/s10958-005-0496-1.

[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.

[12]

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

[13]

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

[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.

[15]

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

[16]

J. Bertoin, Lévy Processes,, Cambridge Tracts in Mathematics, (1996).

[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.

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[33]

X. Cabré and J. M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations,, Comm. Math. Phys., 320 (2013), 679. doi: 10.1007/s00220-013-1682-5.

[34]

L. A. Caffarelli, Further regularity for the Signorini problem,, Comm. Partial Differential Equations, 4 (1979), 1067. doi: 10.1080/03605307908820119.

[35]

L. Caffarelli, C.-H. Chan and A. Vasseur, Regularity theory for nonlinear integral operators,, J. Amer. Math. Soc., 24 (2011), 849. doi: 10.1090/S0894-0347-2011-00698-X.

[36]

L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian,, Invent. Math., 171 (2008), 425. doi: 10.1007/s00222-007-0086-6.

[37]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306.

[38]

L. A. Caffarelli, F. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow,, J. Eur. Math. Soc. (JEMS), 15 (2013), 1701. doi: 10.4171/JEMS/401.

[39]

L. A. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure,, Arch. Rational Mech. Anal., 202 (2011), 537. doi: 10.1007/s00205-011-0420-4.

[40]

L. A. Caffarelli and J. L. Vázquez, Asymptotic behaviour of a porous medium equation with fractional diffusion,, Discrete Cont. Dyn. Systems-A, 29 (2011), 1393.

[41]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann. of Math., 171 (2010), 1903. doi: 10.4007/annals.2010.171.1903.

[42]

A. Capella, J. Davila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non local semilinear equations,, Comm. Partial Diff. Eq., 36 (2011), 1353. doi: 10.1080/03605302.2011.562954.

[43]

J. A. Carrillo, Y. Huang and J. L. Vazquez, in, preparation., ().

[44]

Z. Q. Chen, P. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian,, J. Eur. Math. Soc. (JEMS), 12 (2010), 1307. doi: 10.4171/JEMS/231.

[45]

S. Cifani and E. R. Jakobsen, Entropy solution theory for fractional degenerate convection-diffusion equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 413. doi: 10.1016/j.anihpc.2011.02.006.

[46]

S. Cifani, E. R. Jakobsen and K. H. Karlsen, The discontinuous Galerkin method for fractional degenerate convection-diffusion equations,, BIT, 51 (2011), 809. doi: 10.1007/s10543-011-0327-3.

[47]

J. S. Chapman, J. Rubinstein and M. Schatzman, A mean-field model for superconducting vortices,, Eur. J. Appl. Math., 7 (1996), 97. doi: 10.1017/S0956792500002242.

[48]

R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC, (2004).

[49]

E. B. Davies, Heat Kernels and Spectral Theory,, Cambridge Tracts in Mathematics, (1990).

[50]

A. de Pablo and J. L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation,, J. Differential Equations, 93 (1991), 19. doi: 10.1016/0022-0396(91)90021-Z.

[51]

A. De Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A fractional porous medium equation,, Advances in Mathematics, 226 (2011), 1378. doi: 10.1016/j.aim.2010.07.017.

[52]

A. De Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A general fractional porous medium equation,, Comm. Pure Appl. Math., 65 (2012), 1242. doi: 10.1002/cpa.21408.

[53]

A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, Classical solutions for a logarithmic fractional diffusion equation,, to appear in Journal de Math. Pures Appliquées, ().

[54]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces,, preprint, (2011).

[55]

G. Duvaut and J.-L. Lions, Les Inéquations en Mechanique et en Physique,, Travaux et Recherches Mathématiques, (1972).

[56]

W. E, Dynamics of vortex-liquids in Ginzburg-Landau theories with applications to superconductivity,, Phys. Rev. B, 50 (1994), 1126.

[57]

R. A. Fisher, The wave of advance of advantagenous genes,, Ann. Eugenics, 7 (1937), 355.

[58]

R. K. Getoor, First passage times for symmetric stable processes in space,, Trans. Amer. Math. Soc., 101 (1961), 75. doi: 10.1090/S0002-9947-1961-0137148-5.

[59]

A. K. Head, Dislocation group dynamics II. Similarity solutions of the continuum approximation,, Phil. Mag., 26 (1972), 65.

[60]

Y. H. Huang, Explicit barenblatt profiles for fractional porous medium equations,, preprint, (2013).

[61]

M. D. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps,, Comm. Pure Applied Math., 62 (2009), 198. doi: 10.1002/cpa.20253.

[62]

M. D. Jara, T. Komorowski and S. Olla, Limit theorems for additive functionals of a Markov chain,, Ann. Appl. Probab., 19 (2009), 2270. doi: 10.1214/09-AAP610.

[63]

M. Jara, Hydrodynamic limit Of particle systems with long jumps,, , ().

[64]

M. Jara, C. Landim and S. Sethuraman, Nonequilibrium fluctuations for a tagged particle in mean-zero one-dimensional zero-range processes,, Probab. Theory Relat. Fields, 145 (2009), 565. doi: 10.1007/s00440-008-0178-2.

[65]

M. Kassmann, A priori estimates for integro-differential operators with measurable kernels,, Calc. Var., 34 (2009), 1. doi: 10.1007/s00526-008-0173-6.

[66]

J. King and P. McCabe, On the Fisher-KPP equation with fast nonlinear diffusion,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 2529. doi: 10.1098/rspa.2003.1134.

[67]

A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation,, Invent. Math., 167 (2007), 445. doi: 10.1007/s00222-006-0020-3.

[68]

A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Etude de l'équation de diffusion avec accroissement de la quantité de matière, et son application à un problème biologique,, Bjul. Moskowskogo Gos. Univ., 17 (1937), 1.

[69]

N. S. Landkof, Foundations of Modern Potential Theory,, Translated from the Russian by A. P. Doohovskoy, (1972).

[70]

E. K. Lenzi, R. S. Mendes and C. Tsallis, Crossover in diffusion equation: Anomalous and normal behaviors,, Physical Review E, 67 (2003). doi: 10.1103/PhysRevE.67.031104.

[71]

F. H. Lin and P. Zhang, On the hydrodynamic limit of Ginzburg-Landau vortices,, Discrete Cont. Dyn. Systems, 6 (2000), 121. doi: 10.3934/dcds.2000.6.121.

[72]

V. I. Mazya and T. O. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces,, Journal Funct. Anal., 195 (2002), 230. doi: 10.1006/jfan.2002.3955.

[73]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations,, Arch. Ration. Mech. Anal., 199 (2011), 493. doi: 10.1007/s00205-010-0354-2.

[74]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach,, Physics Reports, 339 (2000), 1. doi: 10.1016/S0370-1573(00)00070-3.

[75]

R. H. Nochetto, E. Otarola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis,, , ().

[76]

S. Serfaty and J. L. Vazquez, A mean field equation as limit of nonlinear diffusion with fractional laplacian operators,, Calc. Var. PDEs, (2013). doi: 10.1007/s00526-013-0613-9.

[77]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators,, to appear in Proc. Roy. Soc. Edinburgh Sect. A. Available from: , (): 12.

[78]

A. Signorini, Questioni di elasticità non linearizzata e semilinearizzata,, Rendiconti di Matematica e delle sue Applicazioni, 18 (1959), 95.

[79]

L. E. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional Laplace,, Indiana Univ. Math. J., 55 (2006), 1155. doi: 10.1512/iumj.2006.55.2706.

[80]

L. E. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, Comm. Pure Appl. Math., 60 (2007), 6. doi: 10.1002/cpa.20153.

[81]

D. Stan and J. L. Vázquez, The Fisher-KPP equation with nonlinear fractional diffusion,, submitted, (2013).

[82]

D. Stan, F. del Teso and J. L. Vázquez, Finite and infinite speed of propagation for porous medium equations with fractional pressure,, Comptes Rendus Acad. Sci. Paris, 352 (2014), 123.

[83]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).

[84]

G. Talenti, Elliptic equations and rearrangements,, Ann. Scuola Norm. Sup. (4), 3 (1976), 697.

[85]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl. (4), 110 (1976), 353. doi: 10.1007/BF02418013.

[86]

F. del Teso, Finite difference method for a fractional porous medium equation,, to appear in Calcolo, (2013). doi: 10.1007/s10092-013-0103-7.

[87]

F. del Teso and J. L. Vázquez, Finite difference method for a general fractional porous medium equation,, , ().

[88]

E. Valdinoci, From the long jump random walk to the fractional Laplacian,, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33.

[89]

J. L. Vázquez, Symétrisation pour $u_t=\Delta\varphi(u)$ et applications,, C. R. Acad. Sc. Paris, 295 (1982), 71.

[90]

J. L. Vázquez, Smoothing And Decay Estimates For Nonlinear Diffusion Equations. Equations Of Porous Medium Type,, Oxford Lecture Series in Mathematics and its Applications, (2006). doi: 10.1093/acprof:oso/9780199202973.001.0001.

[91]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs, (2007).

[92]

J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators,, in Nonlinear partial differential equations: the Abel Symposium 2010 (ed. H. Kenneth), (2010), 271.

[93]

J. L. Vázquez, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type,, to appear in J. Europ. Math. Soc.; , (2013).

[94]

J. L. Vázquez, A. de Pablo, F. Quirós and A. Rodríguez, Classical solutions and higher regularity for nonlinear fractional diffusion equations;, , ().

[95]

J. L. Vázquez and B. Volzone, Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type,, to appear in J. Math. Pures Appl.; , ().

[96]

J. L. Vázquez and B. Volzone, Optimal estimates for Fractional Fast diffusion equations,, submitted, ().

[97]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous Diffusion: A tutorial,, in Order and Chaos, (2008).

[98]

H. Weinberger, Symmetrization in Uniformly Elliptic Problems,, in 1962 Studies in Mathematical Analysis and Related Topics, (1962), 424.

[99]

H. Weitzner and G. M. Zaslavsky, Some applications of fractional equations. Chaotic transport and complexity in classical and quantum dynamics,, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273. doi: 10.1016/S1007-5704(03)00049-2.

[100]

W. A. Woyczyński, Lévy processes in the physical sciences,, in Lévy Processes - Theory and Applications, (2001), 241.

[101]

Ya. B. Zel'dovich and A. S. Kompanyeets, Towards a theory of heat conduction with thermal conductivity depending on the temperature,, in Collection of Papers Dedicated to 70th Anniversary of A. F. Ioffe, (1950), 61.

[1]

Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171

[2]

Chenchen Mou. Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2335-2362. doi: 10.3934/cpaa.2015.14.2335

[3]

Tadeusz Kulczycki, Robert Stańczy. Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2581-2591. doi: 10.3934/dcdsb.2014.19.2581

[4]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125

[5]

Dengfeng Lü, Shuangjie Peng. On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3327-3352. doi: 10.3934/dcds.2017141

[6]

Bernd Kawohl, Jiří Horák. On the geometry of the $p$-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040

[7]

Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008

[8]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581

[9]

Kenneth H. Karlsen, Süleyman Ulusoy. On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion. Networks & Heterogeneous Media, 2016, 11 (1) : 181-201. doi: 10.3934/nhm.2016.11.181

[10]

Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168

[11]

Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905

[12]

Benoît Pausader, Walter A. Strauss. Analyticity of the nonlinear scattering operator. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 617-626. doi: 10.3934/dcds.2009.25.617

[13]

De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431

[14]

Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393

[15]

Selma Yildirim Yolcu, Türkay Yolcu. Sharper estimates on the eigenvalues of Dirichlet fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2209-2225. doi: 10.3934/dcds.2015.35.2209

[16]

Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813

[17]

Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154

[18]

Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043

[19]

Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann equations driven by a nonhomogeneous differential operator. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1055-1078. doi: 10.3934/cpaa.2011.10.1055

[20]

Yan Hu. Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 947-964. doi: 10.3934/cpaa.2016.15.947

2016 Impact Factor: 0.781

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (40)

Other articles
by authors

[Back to Top]