某些随机非线性发展方程及无穷维动力系统的研究
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摘要
在数学上,20世纪40年代日本数学家K.It(?)(参考文献[38,39],与此同时还有前苏联数学家Gikhman[30])创立了关于Brown运动的随机微积分理论.之后由于随机分析,随机过程以及微分方程理论等工具的迅速发展和精确描述随机物理现象(诸如来自物理,化学,生物等方面的随机现象)的实际需求,在20世纪90年代,H.Crauel,F.Flandoli,R.Temam,L.Arnold,A.Debouard,A.Debussche,G.Da Prato,J.Zabczyk,等建立了随机非线性发展方程及无穷维随机动力系统的基本概念和理论框架.其中F.Flandoli,H.Crauel等对Burgers方程,Navier-Stokes方程,非线性波动方程,非线性扩散方程证明了整体随机吸引子的存在性.近年来随机微分方程作为交叉性学科有了迅速发展.
本论文主要讨论了几类具有实际物理背景的无穷维随机演化方程.证明了这些随机演化方程解的适定性,并且在整体解存在唯一的前提下,讨论了解的长时间行为.
本论文分为五章:
第一章,介绍了随机微分方程研究进展,给出了有关随机动力系统的一些主要定义和主要结果,并阐述了本文的主要结果.
第二章,分别考虑了带初边值的随机Ginzburg-Landau(GL)方程和广义随机GL方程解的长时间行为,验证了它们存在随机动力系统,并且证明了随机动力系统存在随机吸引子.
第三章,考虑了带初值的弱阻尼,带外力的随机Korteweg-de Vries(KdV)方程,应用能量型的估计式以及KdV方程所具有的色散性质,证明了该问题弱随机吸引子的存在性.
第四章,首先证明了带初值的随机Korteweg-de Vries Benjamin-Ono(KdV-BO)方程解的存在性,然后证明了弱阻尼带外力的随机KdV-BO方程对应的Cauchy问题的弱随机吸引子的存在性.
第五章,考虑了随机长短波方程组,证明了解的局部存在性结果.
本论文主要讨论了几类具有实际物理背景的无穷维随机演化方程.证明了这些随机演化方程解的适定性,并且在整体解存在唯一的前提下,讨论了解的长时间行为.
本论文分为五章:
第一章,介绍了随机微分方程研究进展,给出了有关随机动力系统的一些主要定义和主要结果,并阐述了本文的主要结果.
第二章,分别考虑了带初边值的随机Ginzburg-Landau(GL)方程和广义随机GL方程解的长时间行为,验证了它们存在随机动力系统,并且证明了随机动力系统存在随机吸引子.
第三章,考虑了带初值的弱阻尼,带外力的随机Korteweg-de Vries(KdV)方程,应用能量型的估计式以及KdV方程所具有的色散性质,证明了该问题弱随机吸引子的存在性.
第四章,首先证明了带初值的随机Korteweg-de Vries Benjamin-Ono(KdV-BO)方程解的存在性,然后证明了弱阻尼带外力的随机KdV-BO方程对应的Cauchy问题的弱随机吸引子的存在性.
第五章,考虑了随机长短波方程组,证明了解的局部存在性结果.
引文
[1] L. Arnold, Random dynamical system, Springer Monographs in Mathe-matecs, Springer-Verlag, Berlin, 1998.
[2] D. Bekiranov, T. Ogawa, G. Ponce, On the well-posedness of Benney's interaction equation for short and long waves, Advances Diff. Equations, 1, 919-937, 1996.
[3] D. Bekiranov, T. Ogawa, G. Ponce, Interaction equations for short and long dispersive waves, J. Funct. Anal., 158, 357-388, 1998.
[4] J. Bergh and J. Lofstrom, Interpolation Spaces,Springer-Verlag, 1976.
[5] A. de Bouard, A. Debussche, On the stochastic Korteweg-de Vries equation, J. Funct. Anal., 154, 215-251,1998.
[6] A. de Bouard, A. Debussche, Y. Tsutsumi, White noise driven Korteweg-de Veris equation, J. Funct. Anl., 169, 532-558, 1999.
[7] J. Bourgain, Fourier restriction phenomena for certain lattece subsets and applications to nonlinear evolution equations, part Ⅱ, Geom. funct. Anal., 3, 209-262, 1993.
[8] H. R. Brand, R. J. Deissler, Interaction of localized solutions for subcritical bifurcations, Phys Rev Lett, 63, 2801-2804, 1992.
[9] T. Cazenave, F. B. Weissler, The Cauchy problem for the critical nonlinear Schrodinger equations in H~s, Nonlinear Anal. TMA, 14, 807-836, 1990.
[10] H. Y. Chang, Ch. Lien, S. Sukarto, S. Raychaudhury, J. Hill, E. K. Tsikis, K.E. Lonngren, Propagation of ion-acoustic solitons in a non-quiescent plasma, Plasma Phys. Control. Fusion, 28, 675-681, 1986.
[11] H. Crauel, A. Debussche, F.Flandoli, Random attractors, J. Dyn. Diff. Eq. 9(2), 307-341, 1997.
[12] H. Crauel, F. Flaudoli, Attractors for random dynamical systems, Prob. Th. Rel. Fields 100, 365-393, 1994.
[13] J. L. Daleckii, Differential equations with functional derivatives and stochastic equations for generalized random processe, Dol. Akad. Nauk, SSSR, 166, 1035-1038,1966.
[14] D. A. Dawson, Stochastic evolution equations, Math. Biosci., 154(3-4), 187- 316, 1972.
[15] A. Debussche, Hausdorff dimension of random invariant set, J. Math. Pures Appl., 77, 967-988, 1998.
[16] J. Duan, P. Holme, On the Cauchy problem for a generalized Ginzburg- Landau equation, Nonl. Anal. 22, 1033-1040,1994.
[17] J. Duan, P. Holme, E. S. Titi, Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5, 1303-1314, 1992.
[18] J. Duan , P. Holme, E. S. Titi, Regularity approximation and asymptiotic dynamics for a generalized Ginzburg-Landau equation, Nonlinearity, 6, 915-933, 1993.
[19] L. C. Evans, An introduction to stochastic differential equations, Version 1.2.
[20] K. D. Elworthy, Stochastic differential equations on Manifolds, Cambridge University Press,1982.
[21] F. Flandoli, Some remarks on regularity theory and stochastic flows for parabolic SPDE's, Nr. 135, Pisa: Scuola Normale Superiore, 1992.
[22] W. H. Fleming, Distributed parameter stochastic systems in population biology, Lecture Notes in Economy and Mathematical Systems, No.107, eds. A. Bensoussan and J.L. Lions. Springer-Verlag, 179-191, 1975.
[23] H. Gao, B. Guo, Numbers of determining nodes for a generalized Ginzburg-Landau equation, Prog. Nat. Sci, 5, 636-638, 1995.
[24] H. Gao, B. Guo, Finite dimensional inertial forms for 1D generalized Ginzburg-Landau equation. Sci China Ser A, 12, 1233-1247,1995.
[25] C.W. Gardiner, Handbooks of Stochastic Methods for Physics, Chemistry and Natural Sciences. Springer Series in Synergetics 13, Springer-Verlag, 1983.
[26] J. M. Ghidaglia, B.Heron, Dimension of the attractor associated to the Ginzburg-Landau equation, Phys. D 28, 282-304, 1987.
[27] J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrodinger equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 5(4) 365-405, 1988.
[28] J. M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Diff. Eqs., 74, 369-390, 1988.
[29] J. M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Diff. Eqs., 110, 356-359, 1994.
[30] I. I. Gikhman, A method of constructing random processes, Dokl. Akad. Nauk, SSSR, 58, 961-964, 1946.
[31] J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, I. Compactness Methods, Phys. D 95, 191-228, 1996.
[32] O. Goubet, Ricardo M. S. Rosa, Asympototic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Diff. Eq., 185, 25-53, 2002.
[33] R. Grimshaw, E. Pelinovsky, x. Tian, Interaction of a solitary wave with an external force, Phys. D, 77, 405-433, 1994.
[34] L. Gross, Potential theory in Hilbert spaces, J. Funct. Anal., 1, 123-189, 1965.
[35] B. Guo, Z. Huo, The well-posedness of the Korteweg de Vries-Benjamin-Ono equation, J. Math. Ana. Appl. 295, 444-458, 2004.
[36] R. Herman, The stochastic, damped Korteweg-de Vries equation, J. Phys. A., 23, 1063-1084, 1990.
[37] Z. Huo, B. Guo, Local well-posedness if interaction equations for short and long dispersive waves, J. Partial Diff. Eq., 17, 137-151, 2004.
[38] K. Ito, Differential equations determining Markov processes(in Japanese), Zenkoku Shijo Sugaku Danwakai 1077, 1352-1400, 1942.
[39] K. Ito, Stochastic integral, Proc. Imp. Acad. Tokyo, 20, 519-524, 1944.
[40] G. Jonalasinio, S. K. Mitter, On the stochastic quantization of field theory, Comm. Math. Phys., 101, 409-436, 1985.
[41] R. E. Kalman, A new approach to linear filtering and prediction problems, Trans. ASME Ser. D, J. Basic Eng., 82, 34-45, 1960.
[42] R. E. Kalman, R. S. Bucy, New results in linear filtering and prediction theory, Trans. ASME, Ser. D, J. Basic Eng., 83,95-108,1961.
[43] C. E. Kenig, G. Ponce, L. Vega, A bilinear estimate with application to the Korteweg-De Vries equation, J. of the American Math. Society., 9, 573-603, 1996.
[44] C.E. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ.Math.J., 40, 33-69, 1991.
[45] C.E. Kenig, G. Ponce, L. Vega, On the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc., 342,155-172, 1994.
[46] C. E. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71, 1-21, 1993.
[47] J. U. Kim, On the stochastic Benjamin-Ono equation, J. Differential Equations, 228(2), 737-768, 2006.
[48] V. V. Konotop, L. Vasquez, Nonlinear random waves, World Scientific, Sinapore, 1994.
[49] N. V. Krylov, Controlled diffusion processes (translated from Russian by A. B. Aries), Springer, NY1980.
[50] H. Kunita, Stochastic flows and stochastic differential equations, Cambridge University Press,1990.
[51]P.Langevin,Sur La thèorie du mouvement brownien,C.R.Acad.Sci.Paris,146,530-533,1908.
[52]P.Laurencot,Compact attractor for weakly damped driven Korteweg-de Vries equations on the real line,Czechoslovak Math.J.,48(123),85-94,1998.
[53]F.Linares,L~2 Global well-posedness of the initial value problem associated to the Benjamin equation,J.Diff.Eq.,152,1425-1433,1999.
[54]D.Liu,Convergence of the spectral method for Stochastic Ginzburg-Landau equation by space-time white noise,Comm.Math.Sci,5,361-375,2003.
[55]Y.Matsuno,Stochastic Benjamin-Ono equation and its application to the dynamics of nonlinear random waves,Physical Review E,Volume 54(6),6313-6322,1996.
[56]Y.Matsuno,Forced Benjamin-Ono equation and its application to soliton dynamics,Phys.Rev.E 52,6333 - 6343,1995.
[57]I.Moise and R.Rosa,On the regularity of the global attractor of a weakly damped,forced Korteweg-de Vries equation,Adv.Diff.Eqs.,2,257-296,1997.
[58]I.Moise,R.Rosa,X.Wang,Attractors for non-compact semigroups via energy equations,Nonlinearity,11(5),1369-1393,1998.
[59]E.Pardoux,Stochastic partial differential equations and filtering of diffusion processes,Stochastics,3,127-167,1979.
[60]G.D.Prato,J.Zabczyk,Stochastic Equations in Infinite Dimensions(Encyclopedia of Mathematics and its Applications),Cambridge University Press,1993.
[61]G.D.Prato,A.Debussche,R.Temam,Stochastic Burgers' equation,Nonlinear Differential Equation Appl,1,389-402,1994.
[62]J.C.Robinson,Infinite-Dimensional Dynamical Systems - An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,Cambridge Texts in Applied Mathematics,2001.
[63]R.Rosa,The global attractor of a weakly damped,forced Korteweg-de Vries equation in H~1(R),Mat.Contemp.,19,129-152,2000.
[64]M.Scalerandi,A.Romano,C.A.Condat,Korteweg-de Vries solitons under additive stochastic perturbations,Phys.Rev.E,58,4166-4173,1998.
[65]A.V.Skorohod,Studies in the thoery of random processed,(translated from the Russian by Scripta Technica) Addison Wesley,Reading,Massl,1965.
[66]G.Staffilani,Quadratic forms for a 2-D semilinear Schr(o|¨)dinger equations, Duke Math.J.,86(1),79-107,1997.
[67]D.W.Stroock,S.R.S.Varadhan,Multidimensional diffusion processes,Springer,Berlin,1979.
[68]T.Tao,Multilinear weighted convolution of L~2 functions,and applications to nonlinear dispersive equation,Amer.J.Math.,123,839-908,2001.
[69]R.Temam,Infinite-Dimensional Systems in Mechanics and Physics,Springer-Verlag,New York,1988.
[70]M.Wadati,Stochastic Korteweg-de Vries equation,J.Phys.Soc.Japan 52,2642-2648,1983.
[71]M.Wadati,Y.Akutsu,Stochastic Korteweg-de Vries equation with and without damping,J.Phys.Soc.Japan,53,3342-3350,1984.
[72]D.Yang,The asymptotic behavior of the stochastic Ginzburg-Landau equation with multiplicative noise,J.Math.Phys.45(11),4064-4076,2004.
[73]龚光鲁,随机微分方程引论(第二版),北京大学出版社,2000.
[74]黄志远,随机分析学基础(第二版),科学出版社,2001.
[75]泽夫·司曲斯(Zeev Schuss)(刘永才等译),随机微分方程理论及其应用,上海科学技术文献出版社,1986.
[76]A.费里德曼(吴让泉译),随机微分方程及应用,北京:科学出版社,1983.
[77]郭柏灵,黄海洋,蒋慕蓉,金兹堡-朗道方程,科学出版社,2002.
[78]郭柏灵,高洪俊,广义Ginzburg-Landau方程的有限维行为,自然科学进展,4(4),423-434,1994.
[79]谭绍滨,韩永前,一般非线性发展方程解的长时间行为,数学年刊,16A(2),127-141,1995.
[2] D. Bekiranov, T. Ogawa, G. Ponce, On the well-posedness of Benney's interaction equation for short and long waves, Advances Diff. Equations, 1, 919-937, 1996.
[3] D. Bekiranov, T. Ogawa, G. Ponce, Interaction equations for short and long dispersive waves, J. Funct. Anal., 158, 357-388, 1998.
[4] J. Bergh and J. Lofstrom, Interpolation Spaces,Springer-Verlag, 1976.
[5] A. de Bouard, A. Debussche, On the stochastic Korteweg-de Vries equation, J. Funct. Anal., 154, 215-251,1998.
[6] A. de Bouard, A. Debussche, Y. Tsutsumi, White noise driven Korteweg-de Veris equation, J. Funct. Anl., 169, 532-558, 1999.
[7] J. Bourgain, Fourier restriction phenomena for certain lattece subsets and applications to nonlinear evolution equations, part Ⅱ, Geom. funct. Anal., 3, 209-262, 1993.
[8] H. R. Brand, R. J. Deissler, Interaction of localized solutions for subcritical bifurcations, Phys Rev Lett, 63, 2801-2804, 1992.
[9] T. Cazenave, F. B. Weissler, The Cauchy problem for the critical nonlinear Schrodinger equations in H~s, Nonlinear Anal. TMA, 14, 807-836, 1990.
[10] H. Y. Chang, Ch. Lien, S. Sukarto, S. Raychaudhury, J. Hill, E. K. Tsikis, K.E. Lonngren, Propagation of ion-acoustic solitons in a non-quiescent plasma, Plasma Phys. Control. Fusion, 28, 675-681, 1986.
[11] H. Crauel, A. Debussche, F.Flandoli, Random attractors, J. Dyn. Diff. Eq. 9(2), 307-341, 1997.
[12] H. Crauel, F. Flaudoli, Attractors for random dynamical systems, Prob. Th. Rel. Fields 100, 365-393, 1994.
[13] J. L. Daleckii, Differential equations with functional derivatives and stochastic equations for generalized random processe, Dol. Akad. Nauk, SSSR, 166, 1035-1038,1966.
[14] D. A. Dawson, Stochastic evolution equations, Math. Biosci., 154(3-4), 187- 316, 1972.
[15] A. Debussche, Hausdorff dimension of random invariant set, J. Math. Pures Appl., 77, 967-988, 1998.
[16] J. Duan, P. Holme, On the Cauchy problem for a generalized Ginzburg- Landau equation, Nonl. Anal. 22, 1033-1040,1994.
[17] J. Duan, P. Holme, E. S. Titi, Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5, 1303-1314, 1992.
[18] J. Duan , P. Holme, E. S. Titi, Regularity approximation and asymptiotic dynamics for a generalized Ginzburg-Landau equation, Nonlinearity, 6, 915-933, 1993.
[19] L. C. Evans, An introduction to stochastic differential equations, Version 1.2.
[20] K. D. Elworthy, Stochastic differential equations on Manifolds, Cambridge University Press,1982.
[21] F. Flandoli, Some remarks on regularity theory and stochastic flows for parabolic SPDE's, Nr. 135, Pisa: Scuola Normale Superiore, 1992.
[22] W. H. Fleming, Distributed parameter stochastic systems in population biology, Lecture Notes in Economy and Mathematical Systems, No.107, eds. A. Bensoussan and J.L. Lions. Springer-Verlag, 179-191, 1975.
[23] H. Gao, B. Guo, Numbers of determining nodes for a generalized Ginzburg-Landau equation, Prog. Nat. Sci, 5, 636-638, 1995.
[24] H. Gao, B. Guo, Finite dimensional inertial forms for 1D generalized Ginzburg-Landau equation. Sci China Ser A, 12, 1233-1247,1995.
[25] C.W. Gardiner, Handbooks of Stochastic Methods for Physics, Chemistry and Natural Sciences. Springer Series in Synergetics 13, Springer-Verlag, 1983.
[26] J. M. Ghidaglia, B.Heron, Dimension of the attractor associated to the Ginzburg-Landau equation, Phys. D 28, 282-304, 1987.
[27] J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrodinger equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 5(4) 365-405, 1988.
[28] J. M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Diff. Eqs., 74, 369-390, 1988.
[29] J. M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Diff. Eqs., 110, 356-359, 1994.
[30] I. I. Gikhman, A method of constructing random processes, Dokl. Akad. Nauk, SSSR, 58, 961-964, 1946.
[31] J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, I. Compactness Methods, Phys. D 95, 191-228, 1996.
[32] O. Goubet, Ricardo M. S. Rosa, Asympototic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Diff. Eq., 185, 25-53, 2002.
[33] R. Grimshaw, E. Pelinovsky, x. Tian, Interaction of a solitary wave with an external force, Phys. D, 77, 405-433, 1994.
[34] L. Gross, Potential theory in Hilbert spaces, J. Funct. Anal., 1, 123-189, 1965.
[35] B. Guo, Z. Huo, The well-posedness of the Korteweg de Vries-Benjamin-Ono equation, J. Math. Ana. Appl. 295, 444-458, 2004.
[36] R. Herman, The stochastic, damped Korteweg-de Vries equation, J. Phys. A., 23, 1063-1084, 1990.
[37] Z. Huo, B. Guo, Local well-posedness if interaction equations for short and long dispersive waves, J. Partial Diff. Eq., 17, 137-151, 2004.
[38] K. Ito, Differential equations determining Markov processes(in Japanese), Zenkoku Shijo Sugaku Danwakai 1077, 1352-1400, 1942.
[39] K. Ito, Stochastic integral, Proc. Imp. Acad. Tokyo, 20, 519-524, 1944.
[40] G. Jonalasinio, S. K. Mitter, On the stochastic quantization of field theory, Comm. Math. Phys., 101, 409-436, 1985.
[41] R. E. Kalman, A new approach to linear filtering and prediction problems, Trans. ASME Ser. D, J. Basic Eng., 82, 34-45, 1960.
[42] R. E. Kalman, R. S. Bucy, New results in linear filtering and prediction theory, Trans. ASME, Ser. D, J. Basic Eng., 83,95-108,1961.
[43] C. E. Kenig, G. Ponce, L. Vega, A bilinear estimate with application to the Korteweg-De Vries equation, J. of the American Math. Society., 9, 573-603, 1996.
[44] C.E. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ.Math.J., 40, 33-69, 1991.
[45] C.E. Kenig, G. Ponce, L. Vega, On the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc., 342,155-172, 1994.
[46] C. E. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71, 1-21, 1993.
[47] J. U. Kim, On the stochastic Benjamin-Ono equation, J. Differential Equations, 228(2), 737-768, 2006.
[48] V. V. Konotop, L. Vasquez, Nonlinear random waves, World Scientific, Sinapore, 1994.
[49] N. V. Krylov, Controlled diffusion processes (translated from Russian by A. B. Aries), Springer, NY1980.
[50] H. Kunita, Stochastic flows and stochastic differential equations, Cambridge University Press,1990.
[51]P.Langevin,Sur La thèorie du mouvement brownien,C.R.Acad.Sci.Paris,146,530-533,1908.
[52]P.Laurencot,Compact attractor for weakly damped driven Korteweg-de Vries equations on the real line,Czechoslovak Math.J.,48(123),85-94,1998.
[53]F.Linares,L~2 Global well-posedness of the initial value problem associated to the Benjamin equation,J.Diff.Eq.,152,1425-1433,1999.
[54]D.Liu,Convergence of the spectral method for Stochastic Ginzburg-Landau equation by space-time white noise,Comm.Math.Sci,5,361-375,2003.
[55]Y.Matsuno,Stochastic Benjamin-Ono equation and its application to the dynamics of nonlinear random waves,Physical Review E,Volume 54(6),6313-6322,1996.
[56]Y.Matsuno,Forced Benjamin-Ono equation and its application to soliton dynamics,Phys.Rev.E 52,6333 - 6343,1995.
[57]I.Moise and R.Rosa,On the regularity of the global attractor of a weakly damped,forced Korteweg-de Vries equation,Adv.Diff.Eqs.,2,257-296,1997.
[58]I.Moise,R.Rosa,X.Wang,Attractors for non-compact semigroups via energy equations,Nonlinearity,11(5),1369-1393,1998.
[59]E.Pardoux,Stochastic partial differential equations and filtering of diffusion processes,Stochastics,3,127-167,1979.
[60]G.D.Prato,J.Zabczyk,Stochastic Equations in Infinite Dimensions(Encyclopedia of Mathematics and its Applications),Cambridge University Press,1993.
[61]G.D.Prato,A.Debussche,R.Temam,Stochastic Burgers' equation,Nonlinear Differential Equation Appl,1,389-402,1994.
[62]J.C.Robinson,Infinite-Dimensional Dynamical Systems - An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,Cambridge Texts in Applied Mathematics,2001.
[63]R.Rosa,The global attractor of a weakly damped,forced Korteweg-de Vries equation in H~1(R),Mat.Contemp.,19,129-152,2000.
[64]M.Scalerandi,A.Romano,C.A.Condat,Korteweg-de Vries solitons under additive stochastic perturbations,Phys.Rev.E,58,4166-4173,1998.
[65]A.V.Skorohod,Studies in the thoery of random processed,(translated from the Russian by Scripta Technica) Addison Wesley,Reading,Massl,1965.
[66]G.Staffilani,Quadratic forms for a 2-D semilinear Schr(o|¨)dinger equations, Duke Math.J.,86(1),79-107,1997.
[67]D.W.Stroock,S.R.S.Varadhan,Multidimensional diffusion processes,Springer,Berlin,1979.
[68]T.Tao,Multilinear weighted convolution of L~2 functions,and applications to nonlinear dispersive equation,Amer.J.Math.,123,839-908,2001.
[69]R.Temam,Infinite-Dimensional Systems in Mechanics and Physics,Springer-Verlag,New York,1988.
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