Riv. Mat. Univ. Parma, Vol. 5, No. 1, 2014

Moritz Kassmann [1] and Russell W. Schwab [2]

Regularity results for nonlocal parabolic equations

Pages: 183-212
Received: 14 May 2013   
Accepted in revised form: 8 August 2013
Mathematics Subject Classification (2010): Primary 35B65, Secondary 47G20, 60J75.

Keywords: Integro-differential operator, nonlocal operator, parabolic equation, Moser iteration, weak Harnack inequality, Hölder regularity.
Authors addresses:
[1] : Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
[2] : Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, United States of America

Abstract: We survey recent regularity results for parabolic equations involving nonlocal operators like the fractional Laplacian. We extend the results of [28] and obtain regularity estimates for nonlocal operators with kernels not being absolutely continuous with respect to the Lebesgue measure.


[1] H. Abels and M.Kassmann, The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels, Osaka J. Math. 46 (2009), no. 3, 661-683.
[2] M. T. Barlow, R. F. Bass, Z.-Q. Chen and M. Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc. 361 (2009), no. 4, 1963-1999.
[3] M. T. Barlow, R. F. Bass and T. Kumagai, Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps, Math. Z. 261 (2009), no. 2, 297-320.
[4] M. T. Barlow, A. Grigor’yan and T. Kumagai, Heat kernel upper bounds for jump processes and the first exit time, J. Reine Angew. Math. 626 (2009), 135-157.
[5] R. F. Bass, Regularity results for stable-like operators, J. Funct. Anal. 257 (2009), no. 8, 2693-2722.
[6] R. F. Bass, M. Kassmann and T. Kumagai, Symmetric jump processes: localization, heat kernels and convergence, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 1, 59-71.
[7] R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal. 17 (2002), no. 4, 375-388.
[8] R. F. Bass and D. A. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2933-2953.
[9] K. Bogdan and P. Sztonyk, Harnack’s inequality for stable Lévy processes, Potential Anal. 22 (2005), no. 2, 133-150.
[10] L. Caffarelli, C. H. Chan and A. Vasseur, Regularity theory for parabolic nonlinear integral operators, J. Amer. Math. Soc. 24 (2011), no. 3, 849-869.
[11] Luis Caffarelli and Luis Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 62 (2009), no. 5, 597-638.
[12] L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 59-88.
[13] L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2) 171 (2010), no. 3, 1903-1930.
[14] P. Cardaliaguet and C. Rainer, Hölder regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi equations with superquadratic growth in the gradient, SIAM J. Control Optim. 49 (2011), no. 2, 555-573.
[15] H. Chang Lara and G. Dávila, Regularity for solutions of non local parabolic equations, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 139-172.
[16] Z.-Q. Chen, P. Kim and T. Kumagai, Weighted Poincaré inequality and heat kernel estimates for finite range jump processes, Math. Ann. 342 (2008), no. 4, 833-883.
[17] Z.-Q. Chen, P. Kim and T. Kumagai, Global heat kernel estimates for symmetric jump processes, Trans. Amer. Math. Soc. 363 (2011), no. 9, 5021-5055.
[18] Z.-Q. Chen, P. Kim and R. Song, Dirichlet heat kernel estimates for subordinate brownian motions with Gaussian components, arXiv:1303.6626 [math.PR].
[19] Z.-Q. Chen, P. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1307- 1329.
[20] Z.-Q. Chen, P. Kim and R. Song, Two-sided heat kernel estimates for censored stable-like processes, Probab. Theory Related Fields 146 (2010), no. 3-4, 361-399.
[21] Z.-Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stochastic Process. Appl. 108 (2003), no. 1, 27-62.
[22] Z.-Q. Chen and T. Kumagai, Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields 140 (2008), no. 1, 277-317.
[23] M. G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1-67.
[24] J. Droniou and C. Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal. 182 (2006), no. 2, 299-331.
[25] B. Dyda and M. Kassmann, Regularity estimates for elliptic nonlocal operators, arXiv:1109.6812 [math.AP].
[26] B. Dyda and M. Kassmann, On weighted Poincaré inequalities, Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 2, 721-726.
[27] L. Erdös and H.-T. Yau, Gap universality of generalized wigner and beta-ensembles, arXiv:1211.3786 [math.PR].
[28] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Comm. Partial Differential Equations 38 (2013), no. 9, 1539- 1573.
[29] Christophe Gomez, Radiative transport limit for the random Schrödinger equation with long-range correlations, J. Math. Pures Appl. (9) 98 (2012), no. 3, 295-327.
[30] A. Grigor’yan, J. Hu and K.-S. Lau, Estimates of heat kernels for non-local regular Dirichlet forms, Trans. Amer. Math. Soc., to appear.
[31] A. Grigor’yan and T. Kumagai, On the dichotomy in the heat kernel two sided estimates, In “Analysis on graphs and its applications”, Proc. Sympos. Pure Math., vol. 77, Amer. Math. Soc., Providence, RI, 2008, pages 199-210.
[32] N. Guillen and R. W. Schwab, Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations, Arch. Ration. Mech. Anal. 206 (2012), no. 1, 111-157.
[33] C. Imbert, A non-local regularization of first order Hamilton-Jacobi equations, J. Differential Equations 211 (2005), no. 1, 218-246.
[34] S. Jarohs and T. Weth, Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations, Discrete Contin. Dyn. Syst. 34 (2014), no. 6, 2581-2615.
[35] T. Jin and J. Xiong, A fractional Yamabe flow and some applications, J. Reine Angew. Math., to appear, DOI: 10.1515/crelle-2012-0110.
[36] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations 34 (2009), no. 1, 1-21.
[37] P. Kim and A. Mimica, Green function estimates for subordinate Brownian motions: stable and beyond, Trans. Amer. Math. Soc., to appear, DOI: 10.1090/S0002-9947-2014-06017-0.
[38] P. Kim, R. Song and Z. Vondracek, Potential theory of subordinate Brownian motions revisited, In “Stochastic analysis and applications to finance”, Interdiscip. Math. Sci., vol. 13, World Sci. Publ., Hackensack, NJ, 2012, pages 243-290.
[39] A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (2007), no. 3, 445-453.
[40] T. Komatsu, Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms, Osaka J. Math. 25 (1988), no. 3, 697-728.
[41] T. Komatsu, Uniform estimates for fundamental solutions associated with non-local Dirichlet forms, Osaka J. Math. 32 (1995), no. 4, 833-850.
[42] N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161-175, 239.
[43] Y. Maekawa and H. Miura, Upper bounds for fundamental solutions to non-local diffusion equations with divergence free drift, J. Funct. Anal. 264 (2013), no. 10, 2245-2268.
[44] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Sobolev classes and the martingale problem, J. Differential Equations 256 (2014), no. 4, 1581-1626.
[45] J. Moser, On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727-740.
[46] L. Saloff-Coste, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, vol. 289, Cambridge University Press, Cambridge 2002.
[47] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J. 55 (2006), no. 3, 1155-1174.
[48] L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion, Adv. Math. 226 (2011), no. 2, 2020-2039.
[49] N. S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205-226.
[50] L. Wang, On the regularity theory of fully nonlinear parabolic equations, I, Comm. Pure Appl. Math. 45 (1992), no. 1, 27-76.

Home Riv.Mat.Univ.Parma