具有加性噪声的Boussinesq方程的随机吸引子
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摘要
研究带有加性噪声项的Boussinesq型方程初边值问题的解的长时间动力学行为,首先通过一系列变换,把具有加性噪声项的随机微分方程转化为不具有噪声项的微分方程,由确定性理论得到该方程决定一个随机动力系统,然后利用周盛凡和范小明的方法[1-2]对一类算子进行估计,证明半群存在有界吸收集,且半群是一致渐近紧的,从而得到该半群存在全局吸引子.
The long-time behavior for a type of Boussinesq equation was explored in this paper with additive white noise.Fristly,the stochastic differential equation with additive noise term was transformed into a differential equation without noise by some column transformations.A stochastic dynamical system was obtained from the deterministic theory.Then,we estimated a type of operator which was useful to obtain the existence of absorbing sets and asymptotical compactness for the semigroup on the basis of Zhou and Fan's method1-2.Thus,we can get this semigroup the exist global attractors.
The long-time behavior for a type of Boussinesq equation was explored in this paper with additive white noise.Fristly,the stochastic differential equation with additive noise term was transformed into a differential equation without noise by some column transformations.A stochastic dynamical system was obtained from the deterministic theory.Then,we estimated a type of operator which was useful to obtain the existence of absorbing sets and asymptotical compactness for the semigroup on the basis of Zhou and Fan's method1-2.Thus,we can get this semigroup the exist global attractors.
引文
[1]S Zhou,X Fan.Kernel sections for non-autonomous strongly damped wave equations[J].Journal of Mathematical A-nalysis and Applications,2002,275:850-869.
[2]X Fan,S Zhou.Kernel sections for non-autonomous strongly damped wave equations of non-degenerate Kirchhofftype[J].Applied Mathematics&Computation,2004,158(1):253-266.
[3]Boussinesq J.Theorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal,en communiquant au liquids contenu dans ce canal des vitesses sensiblemend pareilles de la surface an fond[J].Journal de Mathematiques Pures et Appliquees,1872,17(2):55-108.
[4]Bona JL,Sachs RL.Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equations[J].Communications in Mathematical Physics,1998,118:15-29.
[5]Sachs RL.On the blowup of certain solutions of the"good"Boussinesq equations[J].Appl Anal Discr Math,1990(36):145-152.
[6]Deift P,Tomei C,Trubowitz E.Inverse scattering and Bollssinesq equation[J].Communications on Pure Applied Mathematics,1982(35):567-628.
[7]Levine HA,Sleema BD.A note on the non-existence of global solutions of initial bonudary value problems for the Boussinesq equationutt-uxx-3uxxxx=12u2xx[J].Journal of Mathematical Analysis and Applications,1985(107):206-210.
[8]Yang ZJ,chen GW.Blowup of solutions for a class of generalized Boussinesq equations[J].Acta Mathematica Scientia,1996(16):31-40.
[9]李清善,李红,宋长明,等.具阻尼项的“坏”的Boussinesq型方程的cauchy问题[J].北京大学学报(自然科学版),2008,44(5):676-682.
[10]Yang ZJ,Guo BL.Cauchy problem for the multi-dimensional Boussinesq-type equation[J].Journal of Mathematical Analysis and Applications,2008(340):64-80.
[11]Lai SY,Wu YH.The asymptotic solution of the cauchy problem for a generalized Boussinesq equation[J].Discrete Continuous Dynamical System,2003,3(3):401-408.
[12]Hale J.Asympotic Behavior of Dissipative sysrems[M].Providence RI:American Mathematical society,1988.
[13]Teman R.Infinite Dimensional Dynamical system in Mathematics and Physics[M].New York:springer,1997.
[14]Yang XB,zhao CD,Cao J.Dynamics of the discrete coupled nonlinear schrodinger-Boussinesq equation[J].Applied Mathematics&Computation,2013,219(16):8508-8524.
[15]Varlamov V.Existence and uniqueness of a solution to the Cauchy problem for the damped Boussinesq equation[J].Mathematical Methods in the Applied Sciences,1996,19(8):639-649.
[16]Varlamov V.Long-time asymptotics of solutions of the second initial-boundary value problem for the damped Boussinesq equation[J].Abstract and Applied Analysis,1998(2):281-289.
[17]李春秋,赵才地,王玮明.非自治耗散schroinger-Boussinesq方程组紧致核截面的存在性[J].数学年刊(A辑),2014(35):307-322.
[18]Yang ZJ.Longtime dynamics of the damped Boussinesq equation[J].Journal of Mathematical Analysis and Applications,2013(399):180-190.
[19]Chueshov I,Eller M,Lasiecka I.On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation[J].Communications in Partial Differential Equations,2002,27(9-10):1901-1951.
[20]Wu H,Zheng S.Convergence to equilibrium for the damped semilinear wave equation with critical exponent and dissipative boundary condition[J].Quarterly of Applied Mathematics,2006,64(1):167-188.
[21]Yassine H.Existence and Asymptotic Behavior of Solutions to Semilinear Wave Equations with Nonlinear Damping and Dynamical Boundary Condition[J].Journal of Dynamics&Differential Equations,2012,24(3):645-661.
[22]Barbu V.Nonlinear Differential Equations of Monotone Types in Banach Spaces[J].Springer Monographs in Mathematics,2010,24(3):317.
[23]Vesentini E.Semigroups of Linear Operators and Applications to Partial Differential Equations[J].Applied Mathematical Sciences,1983,44(3):343-369.
[24]A.Pazy,Semigroups of linear operators and applications to partial differential equation[M].Springe-Verlag,New York,1983.
[2]X Fan,S Zhou.Kernel sections for non-autonomous strongly damped wave equations of non-degenerate Kirchhofftype[J].Applied Mathematics&Computation,2004,158(1):253-266.
[3]Boussinesq J.Theorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal,en communiquant au liquids contenu dans ce canal des vitesses sensiblemend pareilles de la surface an fond[J].Journal de Mathematiques Pures et Appliquees,1872,17(2):55-108.
[4]Bona JL,Sachs RL.Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equations[J].Communications in Mathematical Physics,1998,118:15-29.
[5]Sachs RL.On the blowup of certain solutions of the"good"Boussinesq equations[J].Appl Anal Discr Math,1990(36):145-152.
[6]Deift P,Tomei C,Trubowitz E.Inverse scattering and Bollssinesq equation[J].Communications on Pure Applied Mathematics,1982(35):567-628.
[7]Levine HA,Sleema BD.A note on the non-existence of global solutions of initial bonudary value problems for the Boussinesq equationutt-uxx-3uxxxx=12u2xx[J].Journal of Mathematical Analysis and Applications,1985(107):206-210.
[8]Yang ZJ,chen GW.Blowup of solutions for a class of generalized Boussinesq equations[J].Acta Mathematica Scientia,1996(16):31-40.
[9]李清善,李红,宋长明,等.具阻尼项的“坏”的Boussinesq型方程的cauchy问题[J].北京大学学报(自然科学版),2008,44(5):676-682.
[10]Yang ZJ,Guo BL.Cauchy problem for the multi-dimensional Boussinesq-type equation[J].Journal of Mathematical Analysis and Applications,2008(340):64-80.
[11]Lai SY,Wu YH.The asymptotic solution of the cauchy problem for a generalized Boussinesq equation[J].Discrete Continuous Dynamical System,2003,3(3):401-408.
[12]Hale J.Asympotic Behavior of Dissipative sysrems[M].Providence RI:American Mathematical society,1988.
[13]Teman R.Infinite Dimensional Dynamical system in Mathematics and Physics[M].New York:springer,1997.
[14]Yang XB,zhao CD,Cao J.Dynamics of the discrete coupled nonlinear schrodinger-Boussinesq equation[J].Applied Mathematics&Computation,2013,219(16):8508-8524.
[15]Varlamov V.Existence and uniqueness of a solution to the Cauchy problem for the damped Boussinesq equation[J].Mathematical Methods in the Applied Sciences,1996,19(8):639-649.
[16]Varlamov V.Long-time asymptotics of solutions of the second initial-boundary value problem for the damped Boussinesq equation[J].Abstract and Applied Analysis,1998(2):281-289.
[17]李春秋,赵才地,王玮明.非自治耗散schroinger-Boussinesq方程组紧致核截面的存在性[J].数学年刊(A辑),2014(35):307-322.
[18]Yang ZJ.Longtime dynamics of the damped Boussinesq equation[J].Journal of Mathematical Analysis and Applications,2013(399):180-190.
[19]Chueshov I,Eller M,Lasiecka I.On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation[J].Communications in Partial Differential Equations,2002,27(9-10):1901-1951.
[20]Wu H,Zheng S.Convergence to equilibrium for the damped semilinear wave equation with critical exponent and dissipative boundary condition[J].Quarterly of Applied Mathematics,2006,64(1):167-188.
[21]Yassine H.Existence and Asymptotic Behavior of Solutions to Semilinear Wave Equations with Nonlinear Damping and Dynamical Boundary Condition[J].Journal of Dynamics&Differential Equations,2012,24(3):645-661.
[22]Barbu V.Nonlinear Differential Equations of Monotone Types in Banach Spaces[J].Springer Monographs in Mathematics,2010,24(3):317.
[23]Vesentini E.Semigroups of Linear Operators and Applications to Partial Differential Equations[J].Applied Mathematical Sciences,1983,44(3):343-369.
[24]A.Pazy,Semigroups of linear operators and applications to partial differential equation[M].Springe-Verlag,New York,1983.