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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 |
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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