**Title:** On Equations of Stochastic Fluid Mechanics

**Speaker:** Professor Boris Rozovskii

**Speaker Info:** Director, the Center for Applied Mathematical Science, University of Southern California

**Brief Description:**

**Special Note**: **This is a special analysis and probability seminar**

**Abstract:**

\begin{document} \begin{center} {\Large \textbf{On Equations of Stochastic Fluid Mechanics}} \end{center}\begin{center} {Abstract} \end{center}

This paper is concerned with the fluid dynamics modeled by the stochastic flow \begin{equation*} \left\{ \begin{array}{l} \dot{\eta}\left( t,x\right) =u\left( t,\eta\left( t,x\right) \right) +\sigma\left( t,\eta\left( t,x\right) \right) \circ\dot{W} \\ \\ \eta(0,x)=x \end{array} \right. \end{equation*} where the turbulent term is driven by the white noise $\dot{W}$. The motivation for this setting is to understand the motion of fluid parcels in turbulent and randomly forced fluid flows. Stochastic Euler and Navier-Stokes equations for the undetermined components $u(t,x)$ and $% \sigma(t,x)$ of the spatial velocity field are derived from the first principles. The resulting equations include as particular cases the deterministic Navier-Stokes and Euler equations as well as these equations with stochastic forcing. I will also discuss existence and uniqueness of solution and relations between the Wiener chaos expansion of solutions and statistical moments. \end{document}

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