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

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

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

Kyudong Choi ^1,

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

Abstract
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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.

Keywords: nonlinear partial differential equations, super-critical fractional diffusion., Image processing, nonlocal operators, drift diffusion equations, integro-differential equations.

Mathematics Subject Classification: Primary: 35B45, 45G05, 47G2.

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.

show all references

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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