In this paper, we use forward-backward stochastic differential systems tostudy the solution of two and d dimensional ($d\geq 3$) Navier-Stokes-$\alpha$equation. For the two dimensional Navier-Stokes-$\alpha$ equation with spaceperiodic boundary conditions, we derive the Feynmann-Kac formula associatedwith the vorticity equation and prove the global existence and uniqueness ofthe solution. For the d dimensional ($d\geq 3$) case, we prove the localexistence and uniqueness of the solution in Sobolev space.
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机译:在本文中,我们使用前向-后向随机微分系统研究二维和d维($ d \ geq 3 $)Navier-Stokes-$ \ alpha $方程的解。对于具有空间周期边界条件的二维Navier-Stokes-$ \ alpha $方程,我们推导了与涡度方程关联的Feynmann-Kac公式,并证明了该解的全局存在性和唯一性。对于d维($ d \ geq 3 $)情况,我们证明了Sobolev空间中解的局部存在性和唯一性。
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