In the first part of this paper, uniqueness of strong solution is established for the Vlasov-unsteady Stokes problem in 3D. The second part deals with a semi discrete scheme, which is based on the coupling of discontinuous Galerkin approximations for the Vlasov and the Stokes equations for the 2D problem. The proposed method is both mass and momentum conservative. Based on a special projection and also the Stokes projection, optimal error estimates in the case of smooth compactly supported initial data are derived. Moreover, the generalization of error estimates to 3D problem is also indicated. Finally, based on time splitting algorithm, some numerical experiments are conducted whose results confirm our theoretical findings.

翻译：本文第一部分建立了三维Vlasov-非定常Stokes问题强解的唯一性。第二部分针对二维问题，提出基于Vlasov方程与Stokes方程不连续Galerkin近似耦合的半离散格式。该方法同时满足质量守恒与动量守恒特性。通过特殊投影及Stokes投影技术，在初始数据光滑且具紧支集的条件下，推导了最优误差估计。同时指明了误差估计向三维问题的推广途径。最后基于时间分裂算法开展数值实验，结果验证了理论分析的正确性。