# American Institute of Mathematical Sciences

May  2013, 33(5): 1741-1771. doi: 10.3934/dcds.2013.33.1741

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

 1 Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712, United States

Received  December 2011 Revised  April 2012 Published  December 2012

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.
Citation: Kyudong Choi. Persistence of Hölder continuity for non-local integro-differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1741-1771. doi: 10.3934/dcds.2013.33.1741
##### References:
 [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. [2] 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. [3] P. Benilan and H. Brezis, Solutions faibles d'équations d'évolution dans les espaces de Hilbert, Ann. Inst. Fourier (Grenoble), 22 (1972), 311-329. [4] 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. [5] Luis Caffarelli and Alessio Figalli, Regularity of solutions to the parabolic fractional obstacle problem, preprint, arXiv:1101.5170. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] 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. [15] 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. [16] 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. [17] Bartlomiej Dyda and Moritz Kassmann, Comparability and regularity estimates for symmetric nonlocal dirichlet forms, preprint, arXiv:1109.6812. [18] 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. [19] 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. [20] Guy Gilboa and Stanley Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592. [21] 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. [22] 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. [23] 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. [24] 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. [25] Takashi Komatsu, Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms, Osaka J. Math., 25 (1988), 697-728. [26] Takashi Komatsu, Uniform estimates for fundamental solutions associated with non-local Dirichlet forms, Osaka J. Math., 32 (1995), 833-860. [27] Hitoshi Kumano-go, "Pseudodifferential Operators," MIT Press, Cambridge, Mass., 1981. Translated from the Japanese by the author, Rémi Vaillancourt and Michihiro Nagase. [28] 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. [29] 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. [30] Russell W. Schwab, Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680. doi: 10.1137/080737897. [31] 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. [32] 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. [33] Luis Silvestre, Hölder estimates for advection fractional-diffusion equations, preprint, arXiv:1009.5723. [34] Luis Silvestre, On the differentiability of the solution to an equation with drift and fractional diffusion, preprint, arXiv:1012.2401. [35] Elias M. Stein, "Harmonic Analysis," Princeton University Press, NJ, 1993.

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##### References:
 [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. [2] 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. [3] P. Benilan and H. Brezis, Solutions faibles d'équations d'évolution dans les espaces de Hilbert, Ann. Inst. Fourier (Grenoble), 22 (1972), 311-329. [4] 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. [5] Luis Caffarelli and Alessio Figalli, Regularity of solutions to the parabolic fractional obstacle problem, preprint, arXiv:1101.5170. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] 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. [15] 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. [16] 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. [17] Bartlomiej Dyda and Moritz Kassmann, Comparability and regularity estimates for symmetric nonlocal dirichlet forms, preprint, arXiv:1109.6812. [18] 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. [19] 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. [20] Guy Gilboa and Stanley Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592. [21] 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. [22] 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. [23] 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. [24] 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. [25] Takashi Komatsu, Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms, Osaka J. Math., 25 (1988), 697-728. [26] Takashi Komatsu, Uniform estimates for fundamental solutions associated with non-local Dirichlet forms, Osaka J. Math., 32 (1995), 833-860. [27] Hitoshi Kumano-go, "Pseudodifferential Operators," MIT Press, Cambridge, Mass., 1981. Translated from the Japanese by the author, Rémi Vaillancourt and Michihiro Nagase. [28] 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. [29] 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. [30] Russell W. Schwab, Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680. doi: 10.1137/080737897. [31] 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. [32] 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. [33] Luis Silvestre, Hölder estimates for advection fractional-diffusion equations, preprint, arXiv:1009.5723. [34] Luis Silvestre, On the differentiability of the solution to an equation with drift and fractional diffusion, preprint, arXiv:1012.2401. [35] Elias M. Stein, "Harmonic Analysis," Princeton University Press, NJ, 1993.
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