Qualitative Research Categories Stochastic And Deterministic Nonlinear Partial Differential Equations

As we know, many mathematical models were constructed under the ideal con-ditions. However, due to uncertainty in the modelling and external environment, this modelling could be subject to random fluctuations. Firstly, we consider some im-portant shallow water equations, including:the Camassa-Holm equation, the two-component Dullin-Gottwald-Holm system, the two-component b-famaily system, the Degasperis-Procesi equation and the Ostrovsky equation. Next, we consider the Hartree equation with the randomizated initial value and stochastic potential, respec-tively. At last, wo consier the general stochastic evolution equations with fractional Brownian motion. We investigate some characteristic of solutions, such as the well-posedness, large deviation principle and random attractors. The organization of this thesis is as follows:In Chapter1, some physical background and the preliminaries are given. In Section1, we recall the research progress of the Camassa-Holm equation, the two-component Dullin-Gottwald-Holm system, the two-component b-famaily system, the Degasperis-Procesi equation, the Ostrovsky equation, the Hartree equation and the general evolution equations, and point out the research contents and significance of the thesis. In Section2, we introduce some preliminary which will be used in the the-sis, including stochastic Ito integral, the stochastic integration for fractional Brownian motions defined pathwise and some inequalities.In Chapter2, the well-posedness of the stochastic Camassa-Holm equation with additive noise is proved by regularization method. Firstly, the well-posedness for the regularization equation, i.e., stochastic high-order Camassa-Holm equation is ob-tained by contraction mapping principle, where the key point is the the bilinear es-timates in Bourgain spaces Xs,b,b, b<1/2. Then, the consistency estimates for the solutions of the regularization equation are obtained, from which we can check that the solutions is a Cauchy sequence and the limit is the unique solution of the stochas-tic Camassa-Holm equation. At last, we point out that when initial value satisfies some conditions, the solution for the stochastic Camassa-Holm equation will blow-up at finite time.In Chapter3, we consider the well-posedness of the Dullin-Gottwald-Holm sys- tems by the regularization method in Chapter2, and construct the wave-breaking cri-teria and global well-posedness in the corresponding space. At last, we obtain the solitary-wave solutions of the Dullin-Gottwald-Holm systems.In Chapter4, we consider the influence of stochastic effects over two-component b-family system, i.e., the large deviation principle for the solutions of the stochastic two-component b-family system with multiplicative noise. We will also use the regu-larization method to get the well-posedness of the stochastic two-component b-family system. By weak convergence approach along with stochastic control, we can ob-tained the large deviation principle for the solutions of the stochastic two-component b-family system.In Chapter5, instead of the regularization method used in the previous chapters, we obtained the local well-posedness of the stochastic Degasperis-Procesi equation by the iterative technique. Then, we proved the global existence of the solutions by constructing the precise blow-up scenario.Chapter6is devoted to the long time behavior of the stochastic damped forced Ostrovsky equation. Firstly, the global existence of solution is obtained, and cre-ates a random dynamical system. Then, by the energy inequality, we can obtain the consistency estimates. Since the domain is R, we need to construct the asymptotic compactness of the solution. And the asymptotic compactness is usually proved by a tail-estimates. Here, the asymptotic compactness is checked by splitting the solutions into a decaying part plus a regular part.Chapter7is consists of two parts. In Section1, by the randomization of the initial value raised by Burq and Tzvetkov in [11], we can construct the existence of solutions of the Hartree equation in the supercritical spaces, where the authors in [139] obtained the ill-posedness. So, our result improves those in [11] in some sence. In Section1, we consider the Hartree equations with stochastic potential. By the Strichartz esti-mate and the contraction mapping principle, the local well-posedness of the stochastic Hartree equation in the space Lρ(Ω;C([0,τ]; Hs(Rn))∩L9([0,τ]; Ws’r(Rn))), with is obtained. By studying the evolution of mo-mentum and of the energy of the solution, the global existence and uniqueness for solution of the stochastic Hartree equation in H1(Rn) is proved. The blow-up solution is also discussed by generalizing the variance identity.Chapter8shows the existence of random attractors for a class of stochastic evo- lution equations with fractional Brownian motion. Firstly, we are able to prove the existence of a unique solution in Holder continuous functions space by the contraction mapping principle. Standard arguments then allow us to conclude that this solution is random or more precisely creates a random dynamical system. Next, we consider the existence of the random attractors, while the paths satisfy some specified conditions. Here, to obtain a priori estimates, we have to formulate a special Gronwall lemma for discrete time, then we can check the existence of a pullback attractor for a discrete non-autonomous dynamical system. It is not difficult then to extend the existence of a pullback attractor to the original continuous non-autonomous dynamical system. Fi-nally, we prove that provided that the covariance of the fractional Brownian motion with Hurst parameter H>1/2is small, then there exists a random attractor for the stochastic evolution equations with fractional Brownian motion.