American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. 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. 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. [4] Luis Caffarelli, Chi Hin Chan and Alexis Vasseur, Regularity theory for parabolic nonlinear integral operators,, J. Amer. Math. Soc., 24 (2011), 849. doi: 10.1090/S0894-0347-2011-00698-X. [5] Luis Caffarelli and Alessio Figalli, Regularity of solutions to the parabolic fractional obstacle problem,, preprint, (). [6] Luis Caffarelli and Luis Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00220-009-0756-x. [17] Bartlomiej Dyda and Moritz Kassmann, Comparability and regularity estimates for symmetric nonlocal dirichlet forms,, preprint, (). [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. 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, 64 (1999), 107. [20] Guy Gilboa and Stanley Osher, Nonlocal operators with applications to image processing,, Multiscale Model. Simul., 7 (2008), 1005. 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. 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. 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. 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. 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. [26] Takashi Komatsu, Uniform estimates for fundamental solutions associated with non-local Dirichlet forms,, Osaka J. Math., 32 (1995), 833. [27] Hitoshi Kumano-go, "Pseudodifferential Operators,", MIT Press, (1981). [28] Yifei Lou, Xiaoqun Zhang, Stanley Osher and Andrea Bertozzi, Image recovery via nonlocal operators,, J. Sci. Comput., 42 (2010), 185. 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. 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. 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. 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. doi: 10.1002/cpa.20153. [33] Luis Silvestre, Hölder estimates for advection fractional-diffusion equations,, preprint, (). [34] Luis Silvestre, On the differentiability of the solution to an equation with drift and fractional diffusion,, preprint, (). [35] Elias M. Stein, "Harmonic Analysis,", Princeton University Press, (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. 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. 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. [4] Luis Caffarelli, Chi Hin Chan and Alexis Vasseur, Regularity theory for parabolic nonlinear integral operators,, J. Amer. Math. Soc., 24 (2011), 849. doi: 10.1090/S0894-0347-2011-00698-X. [5] Luis Caffarelli and Alessio Figalli, Regularity of solutions to the parabolic fractional obstacle problem,, preprint, (). [6] Luis Caffarelli and Luis Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00220-009-0756-x. [17] Bartlomiej Dyda and Moritz Kassmann, Comparability and regularity estimates for symmetric nonlocal dirichlet forms,, preprint, (). [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. 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, 64 (1999), 107. [20] Guy Gilboa and Stanley Osher, Nonlocal operators with applications to image processing,, Multiscale Model. Simul., 7 (2008), 1005. 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. 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. 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. 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. 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. [26] Takashi Komatsu, Uniform estimates for fundamental solutions associated with non-local Dirichlet forms,, Osaka J. Math., 32 (1995), 833. [27] Hitoshi Kumano-go, "Pseudodifferential Operators,", MIT Press, (1981). [28] Yifei Lou, Xiaoqun Zhang, Stanley Osher and Andrea Bertozzi, Image recovery via nonlocal operators,, J. Sci. Comput., 42 (2010), 185. 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. 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. 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. 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. doi: 10.1002/cpa.20153. [33] Luis Silvestre, Hölder estimates for advection fractional-diffusion equations,, preprint, (). [34] Luis Silvestre, On the differentiability of the solution to an equation with drift and fractional diffusion,, preprint, (). [35] Elias M. Stein, "Harmonic Analysis,", Princeton University Press, (1993).
 [1] Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065 [2] Eitan Tadmor, Prashant Athavale. Multiscale image representation using novel integro-differential equations. Inverse Problems & Imaging, 2009, 3 (4) : 693-710. doi: 10.3934/ipi.2009.3.693 [3] Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191 [4] Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057 [5] Eduardo Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Conference Publications, 2007, 2007 (Special) : 277-285. doi: 10.3934/proc.2007.2007.277 [6] Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17 [7] Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160 [8] Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541 [9] Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977 [10] Sebti Kerbal, Yang Jiang. General integro-differential equations and optimal controls on Banach spaces. Journal of Industrial & Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119 [11] Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907 [12] Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677 [13] Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703 [14] Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032 [15] Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517 [16] Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069 [17] Xu Chen, Jianping Wan. Integro-differential equations for foreign currency option prices in exponential Lévy models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 529-537. doi: 10.3934/dcdsb.2007.8.529 [18] Jaan Janno, Kairi Kasemets. A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination. Inverse Problems & Imaging, 2009, 3 (1) : 17-41. doi: 10.3934/ipi.2009.3.17 [19] Hermann Brunner. The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays. Communications on Pure & Applied Analysis, 2006, 5 (2) : 261-276. doi: 10.3934/cpaa.2006.5.261 [20] Lorena Bociu, Petronela Radu, Daniel Toundykov. Errata: Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping. Evolution Equations & Control Theory, 2014, 3 (2) : 349-354. doi: 10.3934/eect.2014.3.349

2017 Impact Factor: 1.179