# American Institute of Mathematical Sciences

December  2015, 35(12): 5977-5998. doi: 10.3934/dcds.2015.35.5977

## Schauder estimates for solutions of linear parabolic integro-differential equations

 1 Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, United States 2 School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

Received  April 2014 Revised  August 2014 Published  May 2015

We prove optimal pointwise Schauder estimates in the spatial variables for solutions of linear parabolic integro-differential equations. Optimal Hölder estimates in space-time for those spatial derivatives are also obtained.
Citation: Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977
##### References:
 [1] B. Barrera, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators, and its application to nonlocal minimal surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci., XIII (2014), 609-639. doi: 10.2422/2036-2145.201202_007.  Google Scholar [2] R. F. Bass, Regularity results for stable-like operators, J. Funct. Anal., 257 (2009), 2693-2722. doi: 10.1016/j.jfa.2009.05.012.  Google Scholar [3] A. Brandt, Interior Schauder estimates for parabolic differential- (or difference-) equations via the maximum principle, Israel J. Math., 7 (1969), 254-262. doi: 10.1007/BF02787619.  Google Scholar [4] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2), 130 (1989), 189-213. doi: 10.2307/1971480.  Google Scholar [5] L. A. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274.  Google Scholar [6] L. A. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88. doi: 10.1007/s00205-010-0336-4.  Google Scholar [7] L. A. Caffarelli and L. Silvestre, The Evans-Krylov theorem for non local fully non linear equations, Ann. of Math. (2), 174 (2011), 1163-1187. doi: 10.4007/annals.2011.174.2.9.  Google Scholar [8] L. A. Caffarelli, C. Chan and A. Vasseur, Regularity theory for parabolic nonlinear integral operators, J. Amer. Math. Soc., 24 (2011), 849-869. doi: 10.1090/S0894-0347-2011-00698-X.  Google Scholar [9] H. A. Chang Lara and G. Davila, Regularity for solutions of non local parabolic equations, Calc. Var. Partial Differential Equations, 49 (2014), 139-172. doi: 10.1007/s00526-012-0576-2.  Google Scholar [10] H. A. Chang Lara and G. Davila, Regularity for solutions of non local parabolic equations II, J. Differential Equations, 256 (2014), 130-156. doi: 10.1016/j.jde.2013.08.016.  Google Scholar [11] H. Dong and D. Kim, Schauder estimates for a class of non-local elliptic equations, Discrete Contin. Dyn. Syst., 33 (2013), 2319-2347. doi: 10.3934/dcds.2013.33.2319.  Google Scholar [12] H. Dong and S. Kim, Partial Schauder estimates for second-order elliptic and parabolic equations, Calc. Var. Partial Differential Equations, 40 (2011), 481-500. doi: 10.1007/s00526-010-0348-9.  Google Scholar [13] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Comm. Partial Differential Equations, 38 (2013), 1539-1573. doi: 10.1080/03605302.2013.808211.  Google Scholar [14] M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations, Riv. Math. Univ. Parma (N.S.), 5 (2014), 183-212.  Google Scholar [15] B. F. Knerr, Parabolic interior Schauder estimates by the maximum principle, Arch. Ration. Mech. Anal., 75 (1980), 51-58. doi: 10.1007/BF00284620.  Google Scholar [16] D. Kriventsov, $C^{1,\alpha}$ interior regularity for nonlinear nonlocal elliptic equations with rough kernels, Comm. Partial Differential Equations, 38 (2013), 2081-2106. doi: 10.1080/03605302.2013.831990.  Google Scholar [17] N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients, Comm. Partial Differential Equations, 35 (2010), 1-22. doi: 10.1080/03605300903424700.  Google Scholar [18] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I. 1968.  Google Scholar [19] Y. Y. Li and L. Nirenberg, Estimates for elliptic system from composition material, Comm. Pure Appl. Math., 56 (2003), 892-925. doi: 10.1002/cpa.10079.  Google Scholar [20] G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations. IV. Time irregularity and regularity, Differential Integral Equations, 5 (1992), 1219-1236.  Google Scholar [21] L. Lorenzi, Optimal Schauder estimates for parabolic problems with data measurable with respect to time, SIAM J. Math. Anal., 32 (2000), 588-615. doi: 10.1137/S0036141098342842.  Google Scholar [22] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem, Potential Anal., 40 (2014), 539-563. doi: 10.1007/s11118-013-9359-4.  Google Scholar [23] J. Serra, Regularity for fully nonlinear nonlocal parabolic equations with rough kernels, Calculus of Variations and Partial Differential Equations, arXiv:1401.4521, 2014. doi: 10.1007/s00526-014-0798-6.  Google Scholar [24] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970.  Google Scholar [25] G. Tian and X.-J.Wang, A priori estimates for fully nonlinear parabolic equations, Int. Math. Res. Not. IMRN, (2013), 3857-3877.  Google Scholar

show all references

##### References:
 [1] B. Barrera, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators, and its application to nonlocal minimal surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci., XIII (2014), 609-639. doi: 10.2422/2036-2145.201202_007.  Google Scholar [2] R. F. Bass, Regularity results for stable-like operators, J. Funct. Anal., 257 (2009), 2693-2722. doi: 10.1016/j.jfa.2009.05.012.  Google Scholar [3] A. Brandt, Interior Schauder estimates for parabolic differential- (or difference-) equations via the maximum principle, Israel J. Math., 7 (1969), 254-262. doi: 10.1007/BF02787619.  Google Scholar [4] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2), 130 (1989), 189-213. doi: 10.2307/1971480.  Google Scholar [5] L. A. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274.  Google Scholar [6] L. A. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88. doi: 10.1007/s00205-010-0336-4.  Google Scholar [7] L. A. Caffarelli and L. Silvestre, The Evans-Krylov theorem for non local fully non linear equations, Ann. of Math. (2), 174 (2011), 1163-1187. doi: 10.4007/annals.2011.174.2.9.  Google Scholar [8] L. A. Caffarelli, C. Chan and A. Vasseur, Regularity theory for parabolic nonlinear integral operators, J. Amer. Math. Soc., 24 (2011), 849-869. doi: 10.1090/S0894-0347-2011-00698-X.  Google Scholar [9] H. A. Chang Lara and G. Davila, Regularity for solutions of non local parabolic equations, Calc. Var. Partial Differential Equations, 49 (2014), 139-172. doi: 10.1007/s00526-012-0576-2.  Google Scholar [10] H. A. Chang Lara and G. Davila, Regularity for solutions of non local parabolic equations II, J. Differential Equations, 256 (2014), 130-156. doi: 10.1016/j.jde.2013.08.016.  Google Scholar [11] H. Dong and D. Kim, Schauder estimates for a class of non-local elliptic equations, Discrete Contin. Dyn. Syst., 33 (2013), 2319-2347. doi: 10.3934/dcds.2013.33.2319.  Google Scholar [12] H. Dong and S. Kim, Partial Schauder estimates for second-order elliptic and parabolic equations, Calc. Var. Partial Differential Equations, 40 (2011), 481-500. doi: 10.1007/s00526-010-0348-9.  Google Scholar [13] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Comm. Partial Differential Equations, 38 (2013), 1539-1573. doi: 10.1080/03605302.2013.808211.  Google Scholar [14] M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations, Riv. Math. Univ. Parma (N.S.), 5 (2014), 183-212.  Google Scholar [15] B. F. Knerr, Parabolic interior Schauder estimates by the maximum principle, Arch. Ration. Mech. Anal., 75 (1980), 51-58. doi: 10.1007/BF00284620.  Google Scholar [16] D. Kriventsov, $C^{1,\alpha}$ interior regularity for nonlinear nonlocal elliptic equations with rough kernels, Comm. Partial Differential Equations, 38 (2013), 2081-2106. doi: 10.1080/03605302.2013.831990.  Google Scholar [17] N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients, Comm. Partial Differential Equations, 35 (2010), 1-22. doi: 10.1080/03605300903424700.  Google Scholar [18] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I. 1968.  Google Scholar [19] Y. Y. Li and L. Nirenberg, Estimates for elliptic system from composition material, Comm. Pure Appl. Math., 56 (2003), 892-925. doi: 10.1002/cpa.10079.  Google Scholar [20] G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations. IV. Time irregularity and regularity, Differential Integral Equations, 5 (1992), 1219-1236.  Google Scholar [21] L. Lorenzi, Optimal Schauder estimates for parabolic problems with data measurable with respect to time, SIAM J. Math. Anal., 32 (2000), 588-615. doi: 10.1137/S0036141098342842.  Google Scholar [22] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem, Potential Anal., 40 (2014), 539-563. doi: 10.1007/s11118-013-9359-4.  Google Scholar [23] J. Serra, Regularity for fully nonlinear nonlocal parabolic equations with rough kernels, Calculus of Variations and Partial Differential Equations, arXiv:1401.4521, 2014. doi: 10.1007/s00526-014-0798-6.  Google Scholar [24] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970.  Google Scholar [25] G. Tian and X.-J.Wang, A priori estimates for fully nonlinear parabolic equations, Int. Math. Res. Not. IMRN, (2013), 3857-3877.  Google Scholar
 [1] 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 [2] Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021051 [3] 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 [4] Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053 [5] 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 [6] Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053 [7] 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 [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] 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 [10] 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 [11] Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015 [12] 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 [13] Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control & Related Fields, 2021, 11 (4) : 715-737. doi: 10.3934/mcrf.2020044 [14] Martin Bohner, Osman Tunç. Qualitative analysis of integro-differential equations with variable retardation. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 639-657. doi: 10.3934/dcdsb.2021059 [15] Michel Chipot, Senoussi Guesmia. On a class of integro-differential problems. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1249-1262. doi: 10.3934/cpaa.2010.9.1249 [16] Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677 [17] Paola Loreti, Daniela Sforza. Observability of $N$-dimensional integro-differential systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 745-757. doi: 10.3934/dcdss.2016026 [18] Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069 [19] Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517 [20] Kyudong Choi. Persistence of Hölder continuity for non-local integro-differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1741-1771. doi: 10.3934/dcds.2013.33.1741

2020 Impact Factor: 1.392