Friday, March 22, 2013

1303.5329 (Freddy Delbaen et al.)

Navier-Stokes Equations and Forward-Backward Stochastic Differential

Freddy Delbaen, Jinniao Qiu, Shanjian Tang
In the paper, we consider a special coupled forward-backward stochastic differential system (FBSDS) which is associated to the viscous incompressible Navier-Stokes equation and provides a probabilistic solution to the latter via the Feynman-Kac formula. With a probabilistic method, we first prove the existence and uniqueness of the solution to the FBSDS. Then under the same conditions, we verify that this solution leads to a unique local strong solution to the associated Navier-Stokes equation, and for both cases of the small Reynolds number and dimension two, we further give the global strong solutions. Our probabilistic representation formula for the solution to the Navier-Stokes equation involves neither integral nor gradient operators, and it can serve to generate a Monte Carlo solution to the viscous incompressible Navier-Stokes equation.
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