The Navier-Stokes-$α$ equation via forward-backward stochastic differential systems

In this paper, we use forward-backward stochastic differential systems to study the solution of two and d dimensional ($d\geq 3$) Navier-Stokes-$α$ equation. For the two dimensional Navier-Stokes-$α$ equation with space periodic boundary conditions, we derive the Feynmann-Kac formula associated with the vorticity equation and prove the global existence and uniqueness of the solution. For the d dimensional ($d\geq 3$) case, we prove the local existence and uniqueness of the solution in Sobolev space.

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