二维环面上带约束的非自治二维随机Navier-Stokes方程的鞅解

Martingale Solutions of Nonautonomous 2DStochastic Navier-Stokes Equations with Constraint on 2D Toru

研究二维环面上带L2-范数约束的非自治随机Navier-Stokes方程鞅解的存在性.首先,利用随机外力中驱动算子的斜对称性和Galerkin近似解序列的一致估计,得到近似解分布的tight紧性.其次,构造新的概率空间及定义于其上的新随机过程,使得近似解于其上收敛到该随机过程.最后,证明所得新随机过程是原方程的解.同时,得到所得鞅解沿样本轨道唯一,进而得到原方程强解的存在性.

In this paper,we study the existence of martingale solutions for two-dimensional nonautonomous stochastic Navier-Stokes equations with a constraint on the L2-norm of the solution.Firstly,the tightness of laws of approximate solutions is obtained by using the skew-symmetry of the driving operator and uniform estimates of the Galerkin approximate solutions sequence.Secondly,a new probability space and the new stochastic processs defined on it are constructed so that the approximate solutions converge to this new stochastic processs.Finally,we prove that the new stochastic processs is the solution of the original equation.At the same time,we obtain that the martingale solution is unique along the sample orbit,and then obtain the existence of the strong solution of the original equation.