Stochastic 2D Navier-Stokes Equations On Time-Dependent Domains

The objective of this PhD thesis is to obtain the existence,uniqueness and large deviation principle for the solutions of stochastic two-dimensional Navier-Stokes equation on time-dependent domains.The first part of the thesis investigates the existence of uniqueness and solutions to the two-dimensional time-varying stochastic Navier-Stokes equation.We use a finitedimensional approximation method to overcome the difficulties caused by the highly nonlinear nature of the Navier-Stokes equation,the time-varying nature of the region,and the effect of random noise.First we transform the two-dimensional Navier-Stokes equation with time-varying regions into a two-dimensional stochastic Navier-Stokes equation on time-fixed regions,but this process does not reduce the complexity of the problem because in this case its spatial norm is still time-dependent and we cannot obtain the corresponding Ito formula and therefore cannot use monotonicity for example to obtain the existence and uniqueness of the solution.We instead obtain the existence and uniqueness of weak solutions in the probabilistic sense by proving the tightness of the distribution of the solutions to a family of finite dimensional approximation equations,and then obtain the existence and uniqueness of strong solutions in the probabilistic sense by the pathwise uniqueness of the solutions and Yamada-Watanabe theorem.The key lies in the proof of the tightness of the finite dimensional approximation,where we construct a sequence of compact sets by approximating the solutions of the equation in finite dimensions so that their probability sizes are in the appropriate range,and then obtain the result of tightness,which leads to the conclusion of the existence and uniqueness of the solutions of the equation.In the second part we prove the Freidlin-Wentzell large deviation principle for the solutions of the two-dimensional stochastic Navier-Stokes equation on time-dependent domain.Here we consider the large deviation principle of Freidlin-Wentzell type,i.e.,small perturbation large deviation.We can obtain an estimate of the large deviation of the solution of the deterministic equation with respect to the solution of the corresponding stochastic differential equation with perturbation when the decreasing perturbation tends to zero.The key is to show that the corresponding perturbed stochastic NavierStokes equation converges weakly to a skeleton equation whose counterpart is determined by the Cameron-Martin space associated with Brownian motion.In this proof,we still have to consider many problems due to the time-varying region of the equation.This conclusion is based on the proof of the existence and uniqueness of the solution in the first part,so we use a method similar to the one used in the proof of the existence and uniqueness of the solution to obtain the corresponding skeleton equation and the Navier-Stokes equation with time-varying region.The existence and uniqueness of the solutions of the Navier-Stokes equations with perturbations and the energy estimates of their finite dimensional approximations are obtained,and then the conclusion that the stochastic Navier-Stokes equations with perturbations converge weakly to the corresponding determined skeleton equations.This leads to the Freidlin-Wentzell type large deviation principle for the solutions of the two-dimensional stochastic Navier-Stokes equations on time-dependent domain.