R. Mikulevicius

Stochastic Navier--Stokes Equations for Turbulent Flows

This paper concerns the fluid dynamics modelled by the stochastic flow \left\{ \begin{array}{l} \boldsymbol{\dot{\eta}}\left( t,x\right) =\boldsymbol{u}\left( t,\boldsymbol{\eta} \left( t,x\right) \right) +\boldsymbol{\sigma}\left( t,\boldsymbol{\eta}\left( t,x\right) \right) \circ\dot{W}, \\ \\ \boldsymbol{\eta}(0,x)=x, \end{array} \right. where the turbulent term is driven by the white noise

Nonlinear Filtering Revisited: A Spectral Approach

\dot{W}

Global L2-solutions of stochastic Navier–Stokes equations

. The motivation for this setting is to understand the motion of fluid parcels in turbulent and randomly forced fluid flows. Stochastic Euler equations for the undetermined components

On unbiased stochastic Navier–Stokes equations

\boldsymbol{u}(t,x)

On Equations of Stochastic Fluid Mechanics

and

Separation of observations and parameters in nonlinear filtering

\boldsymbol{\sigma}(t,x)

Fourier--Hermite Expansions for Nonlinear Filtering

of the spatial velocity field are derived from the first principles. The resulting equations include as particular cases the deterministic and randomly forced counterparts of these equations.In the second part of the paper, we prove the existence and uniqueness of a strong local solution to the stochastic Navier--Stokes equation in

Uniqueness and absolute continuity of weak solutions for parabolic SPDE's

W_{p}^{1}(\boldsymbol{R}^{d}),d >1,p > d. ...

On

The objective of this paper is to develop an approach to nonlinear filtering based on the Cameron--Martin version of Wiener chaos expansion. This approach gives rise to a new numerical scheme for nonlinear filtering. The main feature of this algorithm is that it allows one to separate the computations involving the observations from those dealing only with the system parameters and to shift the latter off-line.

L_{p}

This paper concerns the Cauchy problem in R d for the stochastic Navier-Stokes equation ∂ 1 u = Δu - (u, ⊇)u - ⊇ p + f(u) + [(σ, ⊇)u - ⊇p + g(u)] o W. u(0) = u 0 , div u = 0, driven by white noise W. Under minimal assumptions on regularity of the coefficients and random forces, the existence of a global weak (martingale) solution of the stochastic Navier-Stokes equation is proved. In the two-dimensional case, the existence and pathwise uniqueness of a global strong solution is shown. A Wiener chaos-based criterion for the existence and uniqueness of a strong global solution of the Navier-Stokes equations is established.