The Random Attractor Of The Development Equation On The Change Domain

In this thesis,we study the dynamics of stochastic evolution equation on the vary-ing domains.We mainly discuss two kinds of varying domains:thin domain and expansion domain.Thin domain refers to a domain collapses onto a lower dimensional spatial do-main.At present,there are some results on the thin domain problem in the literature.The specific problems considered in this thesis are to prove the existence and the con-vergence?i.e.upper semi-continuity?of bi-spatial random attractors in regular space when the domain degenerates to the lower dimensional spatial domain.The expansion domain means that a bounded domain extended to an unbound-ed domain.The problem of expansion domain is a new subject in this thesis,which we mainly studies the existence of random attractor and upper semi-continuity when bounded domain attractor approaches to the unbounded domain for stochastic evolu-tion equation defined on a family of expanding domains.More precisely,the main objective and innovations of this thesis are as follows:First,we show the existence of bi-spatial random attractor and the convergence in regular space when the domain collapses to the lower dimensional spatial domain for stochastic reaction-diffusion equation on a thin domain.The measurability for each attractor A_???is the thin direction measure?in L²and L^pspaces are obtained and borrowing the methods of symbolical truncation and spatial split,we prove the attractiveness and compactness of A_?in L^p.Furthermore,we prove that A_?converges to the attractor of lower dimensional domain under the topology of L^p,when measure converges to zero in thin direction.Second,we prove the existence of random attractor both in p-order Lebesgue s-pace and Sobolev space as well as upper semi-continuity under the p-norm for reaction-diffusion equation with a general multiplicative noise defined on a thin domain.Unlike the system with additive noise,we need to make different assumptions about the sys-tem with a general multiplicative noise to obtain different Lusin continuity in sample and uniform estimates of solutions.On the other hand,we further apply the spec-tral decomposition to obtain the uniform estimates and asymptotic compactness of the cocycle in a Sobolev space.Third,we consider both the?L²,H¹?-continuity with respect to the initial da-ta and?L²,H¹?-random attractor of the stochastic reaction-diffusion equation on a thin domain,where the nonlinearity can be decomposed into two functions with?p,q?-growth exponents.By means of a computation method of induction and bootstrap tech-nique,it is shown that the difference of solutions near an initial time is(L²,L^kp-2k+2)-continuously.In particular,by the 2p-2-order integrability,we show the continuity of solution operator from L²to H¹when k=2,which we further show the existence of?L²,H¹?-random attractor of the stochastic reaction-diffusion system.Fourth,taking the stochastic g-Navier-Stokes equation as an example?refer to re-placing?·?gu?=0 by?·u=0 in the usual NS equation?,we establish a theoretical criterion of the existence and approximation of random attractors on the expansion do-main.Roughly speaking,using the expansion and restriction of functions as well as generalizing the energy-equation method,we show that the sequence of expanding co-cycles are weakly equi-continuous and strongly equi-asymptotically compact,which leads to existence of the null-expansion of corresponding random attractor and upper semi-continuity when unbounded-domain attractor is approximated by the family of bounded-domain attractors.