The Study Of Well-Posedness And Dynamics For Some Types Of Stochastic Partial Differential Equations Arising From Fluid Mechanics

Stochastic partial differential equation has been one of most popular topic in mathematical research in the last few years.It is used to depict the phenomena and models with disturbance,uncertainties in the real world,and also has wide appli-cations in many research fields such as physics,chemistry,biology,climate and so on.This thesis,which consists of five chapters,is mainly concerned with the well-posedness and dynamical behaviors for some types of stochastic partial differential equations arising from fluid mechanics.In Chapter 1,the backgrounds of our study are introduced and the related prelim-inaries are presented.We first recall the physical background and research progress of the concerned stochastic partial differential equations,i.e.the generalized Ginzburg-Landau equation,the coupled Kuramoto-Sivashinsky and Ginzburg-Landau equation,and the higher order modified Camass-Holm equation.Then we present some pre-liminaries used throughout this thesis,like the stochastic integration w.r.t.Poisson random measure,random attractor as well as some useful inequalities.In Chapter 2,the well-posedness for the stochastic generalized Ginzburg-Landau equation driven by a multiplicative jump noise is considered.We notice that there is much literature of the well-posedness for the deterministic or the stochastic general-ized Ginzburg-Landau equation driven by white noise(see,e.g.[51-55,69,96,101-103]).Thus in this thesis we aim to extend the well-posedness result to the stochastic generalized Ginzburg-Landau equation driven by the jump noise.In the proof,we borrow some ideas from Brzezniak and Liu[15],in which the nonlinear term of the concerned stochastic partial differential equation driven by jump noise satisfies the lo-cally monotonic condition.Brzezniak and Liu proved the existence of strong solution of such kind of equation in[15]by the prior estimates,weak convergence and mono-tonicity technique.But we need to point out that,for the generalized Ginzburg-Landau equation,such a locally monotonic condition of the nonlinear term is no longer auto-matically satisfied.For this,we utilize the characteristic structure of nonlinear term and dedicated analysis to cover this gap,which together with Fadedo-Galerkin method and monotonicity technique leads to the existence and uniqueness of the solution.In Chapter 3,the long time behavior of the stochastic Kuramoto-Sivashinsky and Ginzburg-Landau Equations(KS-GL)perturbed by additive noises is investigat-ed.We first obtain the global solution of this coupled system and verify that this system generates a continuous random dynamical system.Then based on the proof for the existence of a compact absorbing set,the proof for the existence of the random attractor of the stochastic KS-GL equation follows.In Chapter 4,the well-posedness for the higher order stochastic modified Camassa-Holm equation driven by zero mean space-time Gaussian process is stud-ied and the limit behavior of the solution as the driven noise decays is demonstrated.For the given initial value u0 ? HS(R)a.s.,where s>-n+5/4 and n>2,we obtain the existence and uniqueness of local solution by establishing a new conser-vation law and bilinear estimates in Bourgain spaces Xs,b(b<1/2).In addition,in the case that u0?L2(?,H1(R))and the driven noise decays,we also prove the existence of solution of stochastic modified Camassa-Holm equation in the space L2(?;C([0,T];H1(R)))which converges to the solution of the corresponding de-terministic modified Camassa-Holm equation.In Chapter 5,the Gibbs measure for the higher order stochastic modified Camassa-Holm equation with randomization on initial data and periodic boundary condition is investigated.It is well known that the Gibbs measure is an efficient tool for some kind of dispersive equations in the continuation of local solution to global solution.It means that once the existence and uniqueness of the local solution on a compact measure-support set is derived,we can obtain the existence of global solution(although it may not be the unique one)by the continuation of local solution.Hence,in some extent,Gibbs measure compensates the absence of conservation law in the Sobolev space with lower regularity.In this chapter,we first construct the Borel mea-sure of Gibbs type in the Sobolev spaces with lower regularity borrowing ideas from[92,93],then we prove the existence of global solution with the help of Prokhorov compactness theorem,Skorokhod convergence theorem and Gibbs measure.