# American Institute of Mathematical Sciences

July  2014, 34(7): 2795-2818. doi: 10.3934/dcds.2014.34.2795

## Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations

 1 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088 2 School of Science, Nanjing University of Science & Technology, Nanjing, 210094, China 3 Department of Mathematics, Nanjing University, Nanjing, 210093, China

Received  October 2012 Revised  September 2013 Published  December 2013

This paper derives a Schrödinger approximation for weakly dissipative stochastic Klein--Gordon--Schrödinger equations with a singular perturbation and scaled small noises on a bounded domain. Detail uniform estimates are given to pass the limit as perturbation and noise disappear. Approximation in two different spaces are considered. Furthermore a large deviation principe of solutions is derived by weak convergence approach.
Citation: Boling Guo, Yan Lv, Wei Wang. Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2795-2818. doi: 10.3934/dcds.2014.34.2795
##### References:
 [1] J. B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Schrödinger-Klein-Gordon equations,, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., (1978), 37. Google Scholar [2] W. Bao, X. Dong and S. Wang, Singular limits of Klein-Gordon-Schrödinger equations to Schrödinger-Yukawa equations,, Mult. Model. Simul., 8 (2010), 1742. doi: 10.1137/100790586. Google Scholar [3] P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling,, SIAM J. Math. Anal., 21 (1990), 1190. doi: 10.1137/0521065. Google Scholar [4] A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems,, Ann. Probab., 36 (2008), 1390. doi: 10.1214/07-AOP362. Google Scholar [5] P. L. Chow, Large deviation problem for some parabolic Itô equations,, Comm. Pure Appl. Math., 45 (1992), 97. doi: 10.1002/cpa.3160450105. Google Scholar [6] J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences,, Stoch. Proc. and Appl., 119 (2009), 2052. doi: 10.1016/j.spa.2008.10.004. Google Scholar [7] P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations,, Wiley Series in Probability and Statistics: Probability and Statistics, (). doi: 10.1002/9781118165904. Google Scholar [8] W. E, X. Li and E. Vanden-Eijnden, Some recent progress in multiscale modeling,, in Multiscale Modelling and Simulation, (2004), 3. doi: 10.1007/978-3-642-18756-8_1. Google Scholar [9] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems,, 2nd edition, (1998). doi: 10.1007/978-1-4612-0611-8. Google Scholar [10] I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations. II,, J. Math. Anal. Appl., 66 (1978), 358. doi: 10.1016/0022-247X(78)90239-1. Google Scholar [11] B. Guo and C. Miao, Asymptotic behavior of coupled Klein-Gordon-Schrödinger equations,, Sci. China Ser. A, 25 (1995), 705. Google Scholar [12] B. Guo and Y. Li, Attractors for Klein-Gordon-Schrödinger equations in $\mathbbR^3$,, J. Diff. Equa., 136 (1997), 356. doi: 10.1006/jdeq.1996.3242. Google Scholar [13] N. Hayashi and W. von Wahl, On the global strong solution of coupled Klein-Gordon-Schrödinger equations,, J. Math. Soc. Japan, 39 (1987), 489. doi: 10.2969/jmsj/03930489. Google Scholar [14] P. Imkeller and A. Monahan, eds., Stochastic climate dynamics,, a special issue in the journal Stoch. and Dyna., 2 (2002). Google Scholar [15] G. Kallianpur and J. Xiong, Large deviations for a class of stochastic partial differential equations,, Ann. Probab., 24 (1996), 320. doi: 10.1214/aop/1042644719. Google Scholar [16] S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation,, Comm. Pure Appl. Math., 38 (1985), 321. doi: 10.1002/cpa.3160380305. Google Scholar [17] Y. Li and B. Guo, Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations,, J. Math. Anal. Appl., 282 (2003), 256. doi: 10.1016/S0022-247X(03)00152-5. Google Scholar [18] K. N. Lu and B. X. Wang, Attractor for dissipative Klein-Gordon-Schrödinger equations in $\mathbbR^3$,, J. Diff. Equa., 170 (2001), 281. Google Scholar [19] Y. Lv, B. Guo and X. Yang, Dynamics of stochastic Klein-Gordon-Schrödinger equations in unbounded domains,, Diff. and Inte. Equa., 24 (2011), 231. Google Scholar [20] Y. Lv and A. J. Roberts, Large deviation principle for singularly perturbed stochastic damped wave equations,, to appear in Stoch. Anal. Appl., (2013). Google Scholar [21] M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces,, Dissertationes Math. (Rozprawy Mat.), 426 (2004). doi: 10.4064/dm426-0-1. Google Scholar [22] T. Ozawa, K. Tsutaya and Y. Tsutsumi, Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions,, Math. Ann., 313 (1999), 127. doi: 10.1007/s002080050254. Google Scholar [23] S. Peszat, Large deviation estimates for stochastic evolution equations,, Probab. Theory Related Fields, 98 (1994), 113. doi: 10.1007/BF01311351. Google Scholar [24] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Encyclopedia of Mathematics and its Applications, (1992). doi: 10.1017/CBO9780511666223. Google Scholar [25] S. H. Schochet and M. I. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence,, Comm. Math. Phys., 106 (1986), 569. doi: 10.1007/BF01463396. Google Scholar [26] R. B. Sowers, Large deviations for a reaction-diffusion equation with non-Gaussian perturbation,, Ann. Probab., 20 (1992), 504. doi: 10.1214/aop/1176989939. Google Scholar [27] S. S. Sritharan and P. Sundar, Large deviations for two-dimensional Navier-Stokes equations with multiplicative noise,, Stochastic Process. Appl., 116 (2006), 1636. doi: 10.1016/j.spa.2006.04.001. Google Scholar [28] T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis,, CBMS Regional Conference Series in Mathematics, (2006). Google Scholar [29] B. Wang and H. Lange, Attractors for Klein-Gordon-Schrödinger equation,, J. Math. Phys., 40 (1999), 2445. doi: 10.1063/1.532875. Google Scholar [30] W. Wang, J. Duan and A. J. Roberts, Large deviations for slow-fast stochastic reaction-diffusion equations,, J. Diff. Equa., 253 (2012), 3501. doi: 10.1016/j.jde.2012.08.041. Google Scholar [31] W. Wang and J. Duan, Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions,, Stoch. Anal. Appl., 27 (2009), 431. doi: 10.1080/07362990802679166. Google Scholar

show all references

##### References:
 [1] J. B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Schrödinger-Klein-Gordon equations,, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., (1978), 37. Google Scholar [2] W. Bao, X. Dong and S. Wang, Singular limits of Klein-Gordon-Schrödinger equations to Schrödinger-Yukawa equations,, Mult. Model. Simul., 8 (2010), 1742. doi: 10.1137/100790586. Google Scholar [3] P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling,, SIAM J. Math. Anal., 21 (1990), 1190. doi: 10.1137/0521065. Google Scholar [4] A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems,, Ann. Probab., 36 (2008), 1390. doi: 10.1214/07-AOP362. Google Scholar [5] P. L. Chow, Large deviation problem for some parabolic Itô equations,, Comm. Pure Appl. Math., 45 (1992), 97. doi: 10.1002/cpa.3160450105. Google Scholar [6] J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences,, Stoch. Proc. and Appl., 119 (2009), 2052. doi: 10.1016/j.spa.2008.10.004. Google Scholar [7] P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations,, Wiley Series in Probability and Statistics: Probability and Statistics, (). doi: 10.1002/9781118165904. Google Scholar [8] W. E, X. Li and E. Vanden-Eijnden, Some recent progress in multiscale modeling,, in Multiscale Modelling and Simulation, (2004), 3. doi: 10.1007/978-3-642-18756-8_1. Google Scholar [9] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems,, 2nd edition, (1998). doi: 10.1007/978-1-4612-0611-8. Google Scholar [10] I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations. II,, J. Math. Anal. Appl., 66 (1978), 358. doi: 10.1016/0022-247X(78)90239-1. Google Scholar [11] B. Guo and C. Miao, Asymptotic behavior of coupled Klein-Gordon-Schrödinger equations,, Sci. China Ser. A, 25 (1995), 705. Google Scholar [12] B. Guo and Y. Li, Attractors for Klein-Gordon-Schrödinger equations in $\mathbbR^3$,, J. Diff. Equa., 136 (1997), 356. doi: 10.1006/jdeq.1996.3242. Google Scholar [13] N. Hayashi and W. von Wahl, On the global strong solution of coupled Klein-Gordon-Schrödinger equations,, J. Math. Soc. Japan, 39 (1987), 489. doi: 10.2969/jmsj/03930489. Google Scholar [14] P. Imkeller and A. Monahan, eds., Stochastic climate dynamics,, a special issue in the journal Stoch. and Dyna., 2 (2002). Google Scholar [15] G. Kallianpur and J. Xiong, Large deviations for a class of stochastic partial differential equations,, Ann. Probab., 24 (1996), 320. doi: 10.1214/aop/1042644719. Google Scholar [16] S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation,, Comm. Pure Appl. Math., 38 (1985), 321. doi: 10.1002/cpa.3160380305. Google Scholar [17] Y. Li and B. Guo, Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations,, J. Math. Anal. Appl., 282 (2003), 256. doi: 10.1016/S0022-247X(03)00152-5. Google Scholar [18] K. N. Lu and B. X. Wang, Attractor for dissipative Klein-Gordon-Schrödinger equations in $\mathbbR^3$,, J. Diff. Equa., 170 (2001), 281. Google Scholar [19] Y. Lv, B. Guo and X. Yang, Dynamics of stochastic Klein-Gordon-Schrödinger equations in unbounded domains,, Diff. and Inte. Equa., 24 (2011), 231. Google Scholar [20] Y. Lv and A. J. Roberts, Large deviation principle for singularly perturbed stochastic damped wave equations,, to appear in Stoch. Anal. Appl., (2013). Google Scholar [21] M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces,, Dissertationes Math. (Rozprawy Mat.), 426 (2004). doi: 10.4064/dm426-0-1. Google Scholar [22] T. Ozawa, K. Tsutaya and Y. Tsutsumi, Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions,, Math. Ann., 313 (1999), 127. doi: 10.1007/s002080050254. Google Scholar [23] S. Peszat, Large deviation estimates for stochastic evolution equations,, Probab. Theory Related Fields, 98 (1994), 113. doi: 10.1007/BF01311351. Google Scholar [24] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Encyclopedia of Mathematics and its Applications, (1992). doi: 10.1017/CBO9780511666223. Google Scholar [25] S. H. Schochet and M. I. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence,, Comm. Math. Phys., 106 (1986), 569. doi: 10.1007/BF01463396. Google Scholar [26] R. B. Sowers, Large deviations for a reaction-diffusion equation with non-Gaussian perturbation,, Ann. Probab., 20 (1992), 504. doi: 10.1214/aop/1176989939. Google Scholar [27] S. S. Sritharan and P. Sundar, Large deviations for two-dimensional Navier-Stokes equations with multiplicative noise,, Stochastic Process. Appl., 116 (2006), 1636. doi: 10.1016/j.spa.2006.04.001. Google Scholar [28] T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis,, CBMS Regional Conference Series in Mathematics, (2006). Google Scholar [29] B. Wang and H. Lange, Attractors for Klein-Gordon-Schrödinger equation,, J. Math. Phys., 40 (1999), 2445. doi: 10.1063/1.532875. Google Scholar [30] W. Wang, J. Duan and A. J. Roberts, Large deviations for slow-fast stochastic reaction-diffusion equations,, J. Diff. Equa., 253 (2012), 3501. doi: 10.1016/j.jde.2012.08.041. Google Scholar [31] W. Wang and J. Duan, Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions,, Stoch. Anal. Appl., 27 (2009), 431. doi: 10.1080/07362990802679166. Google Scholar
 [1] P. D'Ancona. On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 609-640. doi: 10.3934/cpaa.2020029 [2] Ji Shu. Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077 [3] A. F. Almeida, M. M. Cavalcanti, J. P. Zanchetta. Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2039-2061. doi: 10.3934/cpaa.2018097 [4] Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067 [5] Caidi Zhao, Gang Xue, Grzegorz Łukaszewicz. Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4021-4044. doi: 10.3934/dcdsb.2018122 [6] Adriana Flores de Almeida, Marcelo Moreira Cavalcanti, Janaina Pedroso Zanchetta. Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Evolution Equations & Control Theory, 2019, 8 (4) : 847-865. doi: 10.3934/eect.2019041 [7] Peng Gao, Yong Li. Averaging principle for the Schrödinger equations†. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2147-2168. doi: 10.3934/dcdsb.2017089 [8] Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 221-238. doi: 10.3934/dcds.2011.31.221 [9] Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55 [10] Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233 [11] Marilena N. Poulou, Nikolaos M. Stavrakakis. Finite dimensionality of a Klein-Gordon-Schrödinger type system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 149-161. doi: 10.3934/dcdss.2009.2.149 [12] Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221 [13] Fábio Natali, Ademir Pastor. Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system. Communications on Pure & Applied Analysis, 2010, 9 (2) : 413-430. doi: 10.3934/cpaa.2010.9.413 [14] Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040 [15] Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081 [16] Salah Missaoui, Ezzeddine Zahrouni. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$. Communications on Pure & Applied Analysis, 2015, 14 (2) : 695-716. doi: 10.3934/cpaa.2015.14.695 [17] Olivier Goubet, Marilena N. Poulou. Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1525-1539. doi: 10.3934/cpaa.2014.13.1525 [18] Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048 [19] Pavlos Xanthopoulos, Georgios E. Zouraris. A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 239-263. doi: 10.3934/dcdsb.2008.10.239 [20] E. Compaan, N. Tzirakis. Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $\mathbb{R}^+$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3867-3895. doi: 10.3934/dcds.2019156

2018 Impact Factor: 1.143