【作者】 王炜；

【导师】 张土生； 翟建梁；

【作者基本信息】 中国科学技术大学 ， 概率论与数理统计， 2022， 博士

【摘要】 这篇博士论文得到了区域随时间变化的二维随机Navier-Stokes方程的解的存在唯一性和大偏差原理。该论文第一部分研究了区域随时间变化的二维随机Navier-Stokes方程的解的存在唯一性,利用了有限维逼近的方法来克服由Navier-Stokes方程的高度非线性性,区域随时间变化的特点以及随机噪声所带来的困难。首先将区域随时间变化的二维Navier-Stokes方程转化为固定区域上的二维随机Navier-Stokes方程,但这一变换并没有降低问题的复杂度,因为在固定区域上其空间范数仍依赖于时间。这也导致我们无法得到相应的It?公式,因此无法使用例如单调性的方法来得到解的存在唯一性。于是我们转而通过证明一族有限维逼近方程解的分布的胎紧性的方法,来得到概率意义下弱解的存在性,再结合方程本身解的轨道唯一性,由Yamada-Watanabe定理,进而得到概率意义下强解的存在唯一性。其中证明的关键在于有限维逼近的胎紧性证明,通过有限维逼近方程的解构造了一列紧集,使其概率大小在合适的范围内,进而得到胎紧性的结果,从而得到方程解的存在唯一性的结论。第二部分我们证明了区域随时间变化的二维随机Navier-Stokes方程解的Freidlin-Wentzell大偏差原理。这里考虑的是Freidlin-Wentzell型的大偏差原理,即小扰动大偏差。即可以得到当扰动递减趋于0时,确定型方程的解关于相应的带扰动的随机微分方程的解的大偏差估计。在证明过程中我们使用了小扰动大偏差证明中常用且有效的弱收敛方法,其中的关键是证明相应的带扰动的随机Navier-Stokes方程弱收敛到一个其对应的确定型的骨架方程,其中骨架方程中带有与布朗运动相关的Cameron-Martin空间所带来的新增项。在此证明过程中,仍然要考虑因为方程所在区域随时间移动所带来的诸多问题,且这一结论是建立在第一部分证明解的存在唯一性的基础上得到的,因此证明过程将利用与解的存在唯一性证明中相类似的方法,得到相应的骨架方程以及带扰动的随机Navier-Stokes方程的解的存在唯一性,以及其有限维逼近的能量估计等,进而得到带扰动的随机Navier-Stokes方程弱收敛到相应的确定型的骨架方程的结论,从而得到区域随时间变化的二维随机Navier-Stokes方程解的Freidlin-Wentzell大偏差原理。更多还原

【Abstract】 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.更多还原