Advanced Search
Article Contents
Article Contents

Persistence of Hölder continuity for non-local integro-differential equations

Abstract Related Papers Cited by
  • In this paper, we consider non-local integro-differential equations under certain natural assumptions on the kernel, and obtain persistence of Hölder continuity for their solutions. In other words, we prove that a solution stays in $C^\beta$ for all time if its initial data lies in $C^\beta$. This result has an application for a fully non-linear problem, which is used in the field of image processing. In addition, we show Hölder regularity for solutions of drift diffusion equations with supercritical fractional diffusion under the assumption $b\in L^\infty C^{1-\alpha}$ on the divergent-free drift velocity. The proof is in the spirit of [23] where Kiselev and Nazarov established Hölder continuity of the critical surface quasi-geostrophic (SQG) equation.
    Mathematics Subject Classification: Primary: 35B45, 45G05, 47G20.


    \begin{equation} \\ \end{equation}
  • [1]

    Martin T. Barlow, Richard F. Bass, Zhen-Qing Chen and Moritz Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc., 361 (2009), 1963-1999.doi: 10.1090/S0002-9947-08-04544-3.


    Richard F. Bass and David A. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc., 354 (2002), 2933-2953.doi: 10.1090/S0002-9947-02-02998-7.


    P. Benilan and H. Brezis, Solutions faibles d'équations d'évolution dans les espaces de Hilbert, Ann. Inst. Fourier (Grenoble), 22 (1972), 311-329.


    Luis Caffarelli, Chi Hin Chan and Alexis Vasseur, Regularity theory for parabolic nonlinear integral operators, J. Amer. Math. Soc., 24 (2011), 849-869.doi: 10.1090/S0894-0347-2011-00698-X.


    Luis Caffarelli and Alessio FigalliRegularity of solutions to the parabolic fractional obstacle problem, preprint, arXiv:1101.5170.


    Luis Caffarelli and Luis Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.doi: 10.1002/cpa.20274.


    Luis A. Caffarelli and Alexis Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930.doi: 10.4007/annals.2010.171.1903.


    Dongho Chae, Peter Constantin and Jiahong Wu, Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 202 (2011), 35-62.doi: 10.1007/s00205-011-0411-5.


    Zhen-Qing Chen, Panki Kim and Takashi Kumagai, Global heat kernel estimates for symmetric jump processes, Trans. Amer. Math. Soc., 363 (2011), 5021-5055.doi: 10.1090/S0002-9947-2011-05408-5.


    Peter Constantin, Gautam Iyer and Jiahong Wu, Global regularity for a modified critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 57 (2008), 2681-2692.doi: 10.1512/iumj.2008.57.3629.


    Peter Constantin and Vlad Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321.doi: 10.1007/s00039-012-0172-9.


    Peter Constantin and Jiahong Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 1103-1110.doi: 10.1016/j.anihpc.2007.10.001.


    Peter Constantin and Jiahong Wu, Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 159-180.doi: 10.1016/j.anihpc.2007.10.002.


    Antonio Córdoba and Diego Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528.doi: 10.1007/s00220-004-1055-1.


    Michael Dabkowski, Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation, Geom. Funct. Anal., 21 (2011), 1-13.doi: 10.1007/s00039-011-0108-9.


    Hongjie Dong and Nataša Pavlović, Regularity criteria for the dissipative quasi-geostrophic equations in Hölder spaces, Comm. Math. Phys., 290 (2009), 801-812.doi: 10.1007/s00220-009-0756-x.


    Bartlomiej Dyda and Moritz KassmannComparability and regularity estimates for symmetric nonlocal dirichlet forms, preprint, arXiv:1109.6812.


    Susan Friedlander and Vlad Vicol, Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 283-301.doi: 10.1016/j.anihpc.2011.01.002.


    Giambattista Giacomin, Joel L. Lebowitz and Errico Presutti, Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems, in "Stochastic Partial Differential Equations: Six Perspectives" 64 of Math. Surveys Monogr., Amer. Math. Soc., Providence, RI, (1999), 107-152.


    Guy Gilboa and Stanley Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.doi: 10.1137/070698592.


    Niels Jacob, Alexander Potrykus and Jiang-Lun Wu, Solving a non-linear stochastic pseudo-differential equation of Burgers type, Stochastic Process. Appl., 120 (2010), 2447-2467.doi: 10.1016/j.spa.2010.08.007.


    Moritz Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations, 34 (2009), 1-21.doi: 10.1007/s00526-008-0173-6.


    A. Kiselev and F. Nazarov, A variation on a theme of Caffarelli and Vasseur, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 370 (2010), 58-72. (dedicated to Nina Nikolaevna Uraltseva).doi: 10.1007/s10958-010-9842-z.


    A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.doi: 10.1007/s00222-006-0020-3.


    Takashi Komatsu, Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms, Osaka J. Math., 25 (1988), 697-728.


    Takashi Komatsu, Uniform estimates for fundamental solutions associated with non-local Dirichlet forms, Osaka J. Math., 32 (1995), 833-860.


    Hitoshi Kumano-go, "Pseudodifferential Operators," MIT Press, Cambridge, Mass., 1981. Translated from the Japanese by the author, Rémi Vaillancourt and Michihiro Nagase.


    Yifei Lou, Xiaoqun Zhang, Stanley Osher and Andrea Bertozzi, Image recovery via nonlocal operators, J. Sci. Comput., 42 (2010), 185-197.doi: 10.1007/s10915-009-9320-2.


    Changxing Miao and Liutang Xue, On the regularity of a class of generalized quasi-geostrophic equations, J. Differential Equations, 251 (2011), 2789-2821.doi: 10.1016/j.jde.2011.04.018.


    Russell W. Schwab, Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680.doi: 10.1137/080737897.


    Luis Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174.doi: 10.1512/iumj.2006.55.2706.


    Luis Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.doi: 10.1002/cpa.20153.


    Luis SilvestreHölder estimates for advection fractional-diffusion equations, preprint, arXiv:1009.5723.


    Luis SilvestreOn the differentiability of the solution to an equation with drift and fractional diffusion, preprint, arXiv:1012.2401.


    Elias M. Stein, "Harmonic Analysis," Princeton University Press, NJ, 1993.

  • 加载中

Article Metrics

HTML views() PDF downloads(157) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint