We prove a stability estimate, in a suitable expected value, of the $1$-Wasserstein distance between the solution of the continuity equation under a Sobolev velocity field and a measure obtained by pushing forward Dirac deltas whose centers belong to a partition of the domain by a (sort of) explicit forward Euler method. The main tool is a $L^\infty_t (L^p_x)$ estimate on the difference between the regular Lagrangian flow of the velocity field and an explicitly constructed approximation of such flow. Although our result only gives estimates in expected value, it has the advantage of being easily parallelizable and of not relying on any particular structure on the mesh. At the end, we also provide estimates with a logarithmic Wasserstein distance, already used in other works on this particular problem.

翻译：本文针对索博列夫速度场下的连续性方程，在适当的期望值下证明了$1$-Wasserstein距离的稳定性估计，该距离衡量方程解与通过某种显式向前欧拉方法从区域划分中心点推送狄拉克δ函数所得的测度之间的差异。主要工具是速度场的正则拉格朗日流与其显式构造近似流之间差值的$L^\infty_t (L^p_x)$估计。虽然结果仅给出期望值下的估计，但该方法具有易于并行化且不依赖特定网格结构的优势。最后，我们还提供了采用对数Wasserstein距离的估计——该距离已在其他关于此特定问题的研究中被采用。