Some Nonlinear Evolution Equations Of The Whole Attractor And Asymptotic Behavior

It is well known, Zakharov system is one of the most important nonlinear dynam-ical models in plasma physics, which describes the interaction between high frequency Langmuir waves and low frequency ion-acoustic waves. In the past decades, because it possesses important physical meaning, many mathematicians and physicists have been quite extensively interested in such system, and many significant and heartening results have been obtained in theoretical and numerical aspects. In order to study deeply, many physicists continually modify and improve the classical Zakharov system. They obtained a new important modified Zakharov system model called quantum Zakharov system in2004that explains the interactions between the waves and the particles in plasmas including the quantum effects. This model is more closer to the physical phe-nomenon and the results obtained in experiments than the classical Zakharov system. This thesis mainly investigates the mathematical theories for such nonlinear dynamical systems, and the results obtained here play important roles in the understanding the physical meaning.The dissertation consists of six chapters.In Chapter1, some physical derivations are simply presented for the nonlinear quantum Zakharov system, which will be studied below in this thesis. Besides, a simple description of the main work about this dissertation is also given.In Chapter2, the global well-posedness and the classical limit of the solution for the quantum Zakharov system. By using energy method, on the basis of a priori esti-mates, the global well-posedness of the solution is obtained. Whenhâ†'0(measuring the influence of quantum effects), the usual Zakharov system is the limit system of quan-tum Zakharov system. The solution of the quantum Zakharov converges the solution of the classical Zakharov system under some convergence rate.Chapter3is devoted to obtain the existence of the attractor for quantum Za-kharov system, the estimate of the dimensions of which is also investigated. By the energy method, on the basis of a priori estimates, the existence and uniqueness of the solution for quantum Zakharov system is obtained using classical Galerkin approxima-tion method in difference work spaces. Then the continuous dynamics systems and the semigroup of solutions on different work spaces are formed. Moreover, the exis-tence of attractor is investigated by the theory of the attractor, and the estimate of the Hausdorff and Fractal dimensions are studied.Chapter4concerns the stochastic quantum Zakharov system with random white noise disturbance. on The existence and uniqueness of the solution will be obtained by classical Galerkin approximation method in difference probability spaces, on the basis of a priori estimates in Ito sense. Then the stochastic dynamical system should be obtained in probability spaces. Moreover, the existence of random attractor in the weak topology sense will be obtained.In Chapter5, the existence of the random attractor for classical Zakharov system in the usual topology is investigated, which is stronger than the stochastic Zakharov system in weak topology. The same as the quantum Zakharov system.