We study the 2D Navier-Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time-discretisation showing a convergence rate of order (up to) 1/2. It holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier-Stokes equations was only known with respect to convergence in probability. Our result is based on uniform-in-probability estimates for the continuous as well as the time-discrete solution exploiting the particular structure of the noise.

翻译：我们研究了在周期边界条件下带有输运噪声的二维Navier-Stokes方程。主要结果是对时间离散化误差的估计，表明其收敛阶数可达1/2。该估计在均方误差收敛意义下成立，而此前对于随机Navier-Stokes方程，仅已知在概率收敛意义下存在这样的收敛阶数。这一结果基于对连续解及时间离散解的一致概率估计，并利用了噪声的特殊结构。