Dynamics Of Stochastic Partly Dissipative Reaction-diffusion Equations On The Time-varying Domain

This thesis is devoted to studying the dynamics of stochastic partly dissipative system defined on time-varying domain and the stochastic compressible Navier-Stokes-Korteweg equations.The first part,we study partly dissipative system defined on non-monotone time-varying domains.The uniqueness and existence of the solutions for partly dissipative system are established by using a C~2-diffeomorphism transformation.We also prove the existence of a pullback attractor for the process generated by the weak solution.After that,we establish a new penalty method.Then the uniqueness and existence of the variational solutions for(stochastic)system on monotone time-varying domains are established.The existence of attractor for the process generated by the variational solution is also studied.The other part,we demonstrate the existence of weak and martingale solution to stochastic Navier-Stokes-Korteweg equations on 2D bounded domain.Our approach is semi-deterministic,based on solving the system for each fixed representative of the ran-dom variable and applying an abstract result on measurability of multi-valued maps.Then we prove the existence of weak solution.The mass equation in compressible Navier-Stokes-Korteweg equations is first order hyperbolic equation.Thus,the extremum prin-ciple and classical prior estimate method cannot be used directly.We introduction of arti-ficial viscosity item for the mass equation.Then we prove the tightness and convergence of approximation system.Using Jakubowski-Skorokhod theorem and BDG inequality for Lévy noise,we can prove the existence of martingale solution for compressible Navier-Stokes-Korteweg equation driven by gaussian noise and Lévy noise.