The aim of this work is to understand better the long time behaviour of asymptotically compact random dynamical systems （RDS）, which can be generated by solutions of some stochastic partial differential equations on...The aim of this work is to understand better the long time behaviour of asymptotically compact random dynamical systems （RDS）, which can be generated by solutions of some stochastic partial differential equations on unbounded domains. The conceptual analysis for the long time behavior of RDS will be done through some examples. An application of those analysis will be demonstrated through the proof of the existence of random attractors for asymptotically compact dissipative RDS.展开更多
We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div（o（x）↓△u） and multiplicative noises. Under some mild conditions on the diffusion variable o（x） an...We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div（o（x）↓△u） and multiplicative noises. Under some mild conditions on the diffusion variable o（x） and without any restriction on the upper growth p of nonlinearity, except that p 〉 2, we show the existences of random attractor in D0^1,2（DN, σ） space, where DN is an arbitrary （bounded or unbounded） domain in R^N N 〉 2. For this purpose, some abstract results based on the omega-limit compactness are established.展开更多
We consider random systems generated by two-sided compositions of random surface diffeomorphisms, together with an ergodic Borel probability measure μ. Let D（μω） be its dimension of the sample measure, then we pr...We consider random systems generated by two-sided compositions of random surface diffeomorphisms, together with an ergodic Borel probability measure μ. Let D（μω） be its dimension of the sample measure, then we prove a formula relating D（μω） to the entropy and Lyapunov exponents of the random system, where D （μω） is dimHμω, dimBμm, or dimBμm.展开更多
In this paper, we consider the long time behaviors for the partly dissipative stochastic reaction diffusion equations. The existence of a bounded random absorbing set is firstly discussed for the systems and then an e...In this paper, we consider the long time behaviors for the partly dissipative stochastic reaction diffusion equations. The existence of a bounded random absorbing set is firstly discussed for the systems and then an estimate on the solution is derived when the time is sufficiently large. Then, we establish the asymptotic compactness of the solution operator by giving uniform a priori estimates on the tails of solutions when time is large enough. In the last, we finish the proof of existence a pullback ran- dom attractor in L^2 （R^n） × L^2 （R^n）. We also prove the upper semicontinuity of random attractors when the intensity of noise approaches zero. The long time behaviors are discussed to explain the corresponding physical phenomenon.展开更多
We study the random dynamical system （RDS） generated by the Benald flow problem with multiplicative noise and prove the existence of a compact random attractor for such RDS.
The stochastic generalized Ginzburg-Landau equation with additive noise can be solved pathwise and the unique solution generates a random system. Then we prove the random system possesses a global random attractor in ...The stochastic generalized Ginzburg-Landau equation with additive noise can be solved pathwise and the unique solution generates a random system. Then we prove the random system possesses a global random attractor in H 0 1 .展开更多
The well-posedness and stability properties of a stochastic viscoelastic equation with multiplicative noise,Lipschitz and locally Lipschitz nonlinear terms are investigated.The method of Lyapunov functions is used to ...The well-posedness and stability properties of a stochastic viscoelastic equation with multiplicative noise,Lipschitz and locally Lipschitz nonlinear terms are investigated.The method of Lyapunov functions is used to investigate the asymptotic dynamics when zero is not a solution of the equation by using an appropriate cocycle and random dynamical system.The stability of mild solutions is proved in both cases of Lipschitz and locally Lipschitz nonlinear terms.Furthermore,we investigate the existence of a non-trivial stationary solution which is exponentially stable,by using a general random fixed point theorem for general cocycles.In this case,the stationary solution is generated by the composition of random variable and Wiener shift.In addition,the theory of random dynamical system is used to construct another cocycle and prove the existence of a random fixed point exponentially attracting every path.展开更多
In this paper, the two-layer quasigeostrophic flow model under stochastic wind forcing is considered. It is shown that when the layer depth or density differ- ence across the layers tends to zero, the dynamics on both...In this paper, the two-layer quasigeostrophic flow model under stochastic wind forcing is considered. It is shown that when the layer depth or density differ- ence across the layers tends to zero, the dynamics on both layers synchronizes to an averaged geophysical flow model.展开更多
基金the National NSFC under grant No.50579022the Foundation of Pre-973 Program of China under grant No.2004CCA02500+1 种基金the SRF for the ROCS,SEMthe Talent Recruitment Foundation of HUST
文摘The aim of this work is to understand better the long time behaviour of asymptotically compact random dynamical systems （RDS）, which can be generated by solutions of some stochastic partial differential equations on unbounded domains. The conceptual analysis for the long time behavior of RDS will be done through some examples. An application of those analysis will be demonstrated through the proof of the existence of random attractors for asymptotically compact dissipative RDS.
基金supported by China NSF(11271388)Scientific and Technological Research Program of Chongqing Municipal Education Commission(KJ1400430)Basis and Frontier Research Project of Chongqing(cstc2014jcyj A00035)
文摘We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div（o（x）↓△u） and multiplicative noises. Under some mild conditions on the diffusion variable o（x） and without any restriction on the upper growth p of nonlinearity, except that p 〉 2, we show the existences of random attractor in D0^1,2（DN, σ） space, where DN is an arbitrary （bounded or unbounded） domain in R^N N 〉 2. For this purpose, some abstract results based on the omega-limit compactness are established.
基金Partially supported by NSFC(10571130)NSFC(10501033) and SRFDP of China.
文摘We consider random systems generated by two-sided compositions of random surface diffeomorphisms, together with an ergodic Borel probability measure μ. Let D（μω） be its dimension of the sample measure, then we prove a formula relating D（μω） to the entropy and Lyapunov exponents of the random system, where D （μω） is dimHμω, dimBμm, or dimBμm.
基金This work is supported by The National Natural Science Foundation of China （Grant No： 11301043）. We also express our thanks to the referee for helpful comments and suggestions.
文摘In this paper, we consider the long time behaviors for the partly dissipative stochastic reaction diffusion equations. The existence of a bounded random absorbing set is firstly discussed for the systems and then an estimate on the solution is derived when the time is sufficiently large. Then, we establish the asymptotic compactness of the solution operator by giving uniform a priori estimates on the tails of solutions when time is large enough. In the last, we finish the proof of existence a pullback ran- dom attractor in L^2 （R^n） × L^2 （R^n）. We also prove the upper semicontinuity of random attractors when the intensity of noise approaches zero. The long time behaviors are discussed to explain the corresponding physical phenomenon.
基金Supported by the China Postdoctoral Science Foundation (No. 2005038326)
文摘We study the random dynamical system （RDS） generated by the Benald flow problem with multiplicative noise and prove the existence of a compact random attractor for such RDS.
文摘The stochastic generalized Ginzburg-Landau equation with additive noise can be solved pathwise and the unique solution generates a random system. Then we prove the random system possesses a global random attractor in H 0 1 .
基金supported the visit of research assistant,Jose Nicolas Pina Leon,to the University of Sevilla in 2019.The research of T.Caraballo has been partially supported by Ministerio de Ciencia,Innovacion y Universidades(Spain),FEDER(European Community)under grant PGC2018-096540-B-I00,and P12-FQM-1492(Junta de Andalucia).
文摘The well-posedness and stability properties of a stochastic viscoelastic equation with multiplicative noise,Lipschitz and locally Lipschitz nonlinear terms are investigated.The method of Lyapunov functions is used to investigate the asymptotic dynamics when zero is not a solution of the equation by using an appropriate cocycle and random dynamical system.The stability of mild solutions is proved in both cases of Lipschitz and locally Lipschitz nonlinear terms.Furthermore,we investigate the existence of a non-trivial stationary solution which is exponentially stable,by using a general random fixed point theorem for general cocycles.In this case,the stationary solution is generated by the composition of random variable and Wiener shift.In addition,the theory of random dynamical system is used to construct another cocycle and prove the existence of a random fixed point exponentially attracting every path.
文摘In this paper, the two-layer quasigeostrophic flow model under stochastic wind forcing is considered. It is shown that when the layer depth or density differ- ence across the layers tends to zero, the dynamics on both layers synchronizes to an averaged geophysical flow model.
