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Flows for the stochastic Navier-Stokes equation. (English) Zbl 1012.60056
The purpose of this paper is to transform a stochastic partial differential equation with multiplicative noise into a random partial differential equation. This equation can be solved pathwise without stochastic calculus, and it defines a stochastic flow and a perfect cocycle. As the transformation is bijective, these properties carry over to the stochastic PDE. Moreover, the transformation can be done in a stationary way, in order to investigate ergodic aspects.
The author considers a general class of equations of Navier-Stokes type \[ dX_{s,t}= - AX_{s,t}+B(X_{s,t},X_{s,t})+F(X_{s,t}) +\sum_{j=1}^m C_t^j X_{s,t}\circ dW_{t}^j \] for \(t>s\) with \(X_{s,s}=y\), and the equation is considered to be Stratonovich with real-valued Brownian motions \(W^j\). The mapping \(F\) is Lipschitz, and the linear operators \(C_t^j\) involve derivatives.
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q30 Navier-Stokes equations
37H05 General theory of random and stochastic dynamical systems
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