Global well-posedness and small time asymptotics of stochastic Ladyzhenskaya-Smagorinsky equations with damping on unbounded domains

Abstract

The Ladyzhenskaya-Smagorinsky equations model turbulence phenomena, and are given by $$\frac{\partial \mathbf{u}}{\partial t}-\hat{I}¼\mathrm{div}\left({\left(1+|\nabla \mathbf{u}{|}^2\right)}^{\frac{p-2}{2}}\nabla \mathbf{u}\right)+(\mathbf{u}\cdot \nabla )\mathbf{u}+\nabla p=\mathbf{f},\nabla \cdot \mathbf{u}=0,$$ for $p\ge 2.$ In this work, we consider the stochastic Ladyzhenskaya-Smagorinsky equations with the damping $\hat{I}\pm \mathbf{u}+{\hat{I}}^2|\mathbf{u}{|}^{r-2}\mathbf{u},$ for $r\ge 2$ ($\hat{I}\pm ,{\hat{I}}^2\ge 0$), subjected to multiplicative Gaussian noise in a Poincaré domain (which may be bounded or unbounded) $\mathcal{O}\subset {\mathbb{R}}^d$ ($2\le d\le 4$). We show the local monotonicity ($p\ge \frac{d}{2}+1,r\ge 2$) as well as global monotonicity ($p\ge 2,r\ge 4$) properties of the linear and nonlinear operators, which along with an application of a stochastic version of the Minty-Browder technique imply the existence of a unique pathwise strong solution satisfying the energy equality (Itô formula), which is proved with the help of the methodology developed in [Krylov, \emph{Probab. Theory Related Fields}, {\bf 147} (2010), 583--605.] Then, we discuss the small time asymptotics by studying the effect of small, highly nonlinear, unbounded drifts (small time large deviation principle) for the stochastic Ladyzhenskaya-Smagorinsky equations with damping.