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

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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

Boling Guo ^1, , Yan Lv ^2, and Wei Wang ^3,

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

Abstract
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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.

Keywords: large deviation principle, weak convergence., Yukawa coupling, Klein--Gordon--Schrödinger equations.

Mathematics Subject Classification: Primary: 60H15; Secondary: 65M5.

Citation: Boling Guo, Yan Lv, Wei Wang. Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations. Discrete and Continuous Dynamical Systems, 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., Inst. Mat., Univ. Fed. Rio de Janeiro), North-Holland Math. Stud., 30, North-Holland, Amsterdam-New York, 1978, 37-44.
[2]	W. Bao, X. Dong and S. Wang, Singular limits of Klein-Gordon-Schrödinger equations to Schrödinger-Yukawa equations, Mult. Model. Simul., 8 (2010), 1742-1769. doi: 10.1137/100790586.
[3]	P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212. doi: 10.1137/0521065.
[4]	A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420. doi: 10.1214/07-AOP362.
[5]	P. L. Chow, Large deviation problem for some parabolic Itô equations, Comm. Pure Appl. Math., 45 (1992), 97-120. doi: 10.1002/cpa.3160450105.
[6]	J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences, Stoch. Proc. and Appl., 119 (2009), 2052-2081. doi: 10.1016/j.spa.2008.10.004.
[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, A Wiley-Interscience Publication, J. Wiley & Sons, Inc., New York,1997. doi: 10.1002/9781118165904.
[8]	W. E, X. Li and E. Vanden-Eijnden, Some recent progress in multiscale modeling, in Multiscale Modelling and Simulation, Lect. Notes Comput. Sci. Eng., 39, Springer, Berlin, 2004, 3-21. doi: 10.1007/978-3-642-18756-8_1.
[9]	M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260, Springer-Verlag, New York. 1998. doi: 10.1007/978-1-4612-0611-8.
[10]	I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations. II, J. Math. Anal. Appl., 66 (1978), 358-378. doi: 10.1016/0022-247X(78)90239-1.
[11]	B. Guo and C. Miao, Asymptotic behavior of coupled Klein-Gordon-Schrödinger equations, Sci. China Ser. A, 25 (1995), 705-714.
[12]	B. Guo and Y. Li, Attractors for Klein-Gordon-Schrödinger equations in $\mathbb{R}^3$, J. Diff. Equa., 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242.
[13]	N. Hayashi and W. von Wahl, On the global strong solution of coupled Klein-Gordon-Schrödinger equations, J. Math. Soc. Japan, 39 (1987), 489-497. doi: 10.2969/jmsj/03930489.
[14]	P. Imkeller and A. Monahan, eds., Stochastic climate dynamics, a special issue in the journal Stoch. and Dyna., 2 (2002).
[15]	G. Kallianpur and J. Xiong, Large deviations for a class of stochastic partial differential equations, Ann. Probab., 24 (1996), 320-345. doi: 10.1214/aop/1042644719.
[16]	S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332. doi: 10.1002/cpa.3160380305.
[17]	Y. Li and B. Guo, Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations, J. Math. Anal. Appl., 282 (2003), 256-265. doi: 10.1016/S0022-247X(03)00152-5.
[18]	K. N. Lu and B. X. Wang, Attractor for dissipative Klein-Gordon-Schrödinger equations in $\mathbb{R}^3$, J. Diff. Equa., 170 (2001), 281-316.
[19]	Y. Lv, B. Guo and X. Yang, Dynamics of stochastic Klein-Gordon-Schrödinger equations in unbounded domains, Diff. and Inte. Equa., 24 (2011), 231-260.
[20]	Y. Lv and A. J. Roberts, Large deviation principle for singularly perturbed stochastic damped wave equations, to appear in Stoch. Anal. Appl., 2013.
[21]	M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Math. (Rozprawy Mat.), 426 (2004), 63 pp. doi: 10.4064/dm426-0-1.
[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), 127-140. doi: 10.1007/s002080050254.
[23]	S. Peszat, Large deviation estimates for stochastic evolution equations, Probab. Theory Related Fields, 98 (1994), 113-136. doi: 10.1007/BF01311351.
[24]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.
[25]	S. H. Schochet and M. I. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys., 106 (1986), 569-580. doi: 10.1007/BF01463396.
[26]	R. B. Sowers, Large deviations for a reaction-diffusion equation with non-Gaussian perturbation, Ann. Probab., 20 (1992), 504-537. doi: 10.1214/aop/1176989939.
[27]	S. S. Sritharan and P. Sundar, Large deviations for two-dimensional Navier-Stokes equations with multiplicative noise, Stochastic Process. Appl., 116 (2006), 1636-1659. doi: 10.1016/j.spa.2006.04.001.
[28]	T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.
[29]	B. Wang and H. Lange, Attractors for Klein-Gordon-Schrödinger equation, J. Math. Phys., 40 (1999), 2445-2457. doi: 10.1063/1.532875.
[30]	W. Wang, J. Duan and A. J. Roberts, Large deviations for slow-fast stochastic reaction-diffusion equations, J. Diff. Equa., 253 (2012), 3501-3522. doi: 10.1016/j.jde.2012.08.041.
[31]	W. Wang and J. Duan, Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions, Stoch. Anal. Appl., 27 (2009), 431-459. doi: 10.1080/07362990802679166.

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., Inst. Mat., Univ. Fed. Rio de Janeiro), North-Holland Math. Stud., 30, North-Holland, Amsterdam-New York, 1978, 37-44.
[2]	W. Bao, X. Dong and S. Wang, Singular limits of Klein-Gordon-Schrödinger equations to Schrödinger-Yukawa equations, Mult. Model. Simul., 8 (2010), 1742-1769. doi: 10.1137/100790586.
[3]	P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212. doi: 10.1137/0521065.
[4]	A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420. doi: 10.1214/07-AOP362.
[5]	P. L. Chow, Large deviation problem for some parabolic Itô equations, Comm. Pure Appl. Math., 45 (1992), 97-120. doi: 10.1002/cpa.3160450105.
[6]	J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences, Stoch. Proc. and Appl., 119 (2009), 2052-2081. doi: 10.1016/j.spa.2008.10.004.
[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, A Wiley-Interscience Publication, J. Wiley & Sons, Inc., New York,1997. doi: 10.1002/9781118165904.
[8]	W. E, X. Li and E. Vanden-Eijnden, Some recent progress in multiscale modeling, in Multiscale Modelling and Simulation, Lect. Notes Comput. Sci. Eng., 39, Springer, Berlin, 2004, 3-21. doi: 10.1007/978-3-642-18756-8_1.
[9]	M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260, Springer-Verlag, New York. 1998. doi: 10.1007/978-1-4612-0611-8.
[10]	I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations. II, J. Math. Anal. Appl., 66 (1978), 358-378. doi: 10.1016/0022-247X(78)90239-1.
[11]	B. Guo and C. Miao, Asymptotic behavior of coupled Klein-Gordon-Schrödinger equations, Sci. China Ser. A, 25 (1995), 705-714.
[12]	B. Guo and Y. Li, Attractors for Klein-Gordon-Schrödinger equations in $\mathbb{R}^3$, J. Diff. Equa., 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242.
[13]	N. Hayashi and W. von Wahl, On the global strong solution of coupled Klein-Gordon-Schrödinger equations, J. Math. Soc. Japan, 39 (1987), 489-497. doi: 10.2969/jmsj/03930489.
[14]	P. Imkeller and A. Monahan, eds., Stochastic climate dynamics, a special issue in the journal Stoch. and Dyna., 2 (2002).
[15]	G. Kallianpur and J. Xiong, Large deviations for a class of stochastic partial differential equations, Ann. Probab., 24 (1996), 320-345. doi: 10.1214/aop/1042644719.
[16]	S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332. doi: 10.1002/cpa.3160380305.
[17]	Y. Li and B. Guo, Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations, J. Math. Anal. Appl., 282 (2003), 256-265. doi: 10.1016/S0022-247X(03)00152-5.
[18]	K. N. Lu and B. X. Wang, Attractor for dissipative Klein-Gordon-Schrödinger equations in $\mathbb{R}^3$, J. Diff. Equa., 170 (2001), 281-316.
[19]	Y. Lv, B. Guo and X. Yang, Dynamics of stochastic Klein-Gordon-Schrödinger equations in unbounded domains, Diff. and Inte. Equa., 24 (2011), 231-260.
[20]	Y. Lv and A. J. Roberts, Large deviation principle for singularly perturbed stochastic damped wave equations, to appear in Stoch. Anal. Appl., 2013.
[21]	M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Math. (Rozprawy Mat.), 426 (2004), 63 pp. doi: 10.4064/dm426-0-1.
[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), 127-140. doi: 10.1007/s002080050254.
[23]	S. Peszat, Large deviation estimates for stochastic evolution equations, Probab. Theory Related Fields, 98 (1994), 113-136. doi: 10.1007/BF01311351.
[24]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.
[25]	S. H. Schochet and M. I. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys., 106 (1986), 569-580. doi: 10.1007/BF01463396.
[26]	R. B. Sowers, Large deviations for a reaction-diffusion equation with non-Gaussian perturbation, Ann. Probab., 20 (1992), 504-537. doi: 10.1214/aop/1176989939.
[27]	S. S. Sritharan and P. Sundar, Large deviations for two-dimensional Navier-Stokes equations with multiplicative noise, Stochastic Process. Appl., 116 (2006), 1636-1659. doi: 10.1016/j.spa.2006.04.001.
[28]	T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.
[29]	B. Wang and H. Lange, Attractors for Klein-Gordon-Schrödinger equation, J. Math. Phys., 40 (1999), 2445-2457. doi: 10.1063/1.532875.
[30]	W. Wang, J. Duan and A. J. Roberts, Large deviations for slow-fast stochastic reaction-diffusion equations, J. Diff. Equa., 253 (2012), 3501-3522. doi: 10.1016/j.jde.2012.08.041.
[31]	W. Wang and J. Duan, Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions, Stoch. Anal. Appl., 27 (2009), 431-459. doi: 10.1080/07362990802679166.

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