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 $$\frac{\partial \boldsymbol{u}}{\partial t}-\mu \mathrm{div}\left(\left(1+|\nabla\boldsymbol{u}|^2\right)^{\frac{p-2}{2}}\nabla\boldsymbol{u}\right)+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\nabla p=\boldsymbol{f}, \ \nabla\cdot\boldsymbol{u}=0,$$ for $p\geq 2.$ In this work, we consider the stochastic Ladyzhenskaya-Smagorinsky equations with the damping $\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-2}\boldsymbol{u},$ for $r\geq 2$ ($\alpha,\beta\geq 0$), subjected to multiplicative Gaussian noise in a Poincar\'e domain (which may be bounded or unbounded) $\mathcal{O}\subset\mathbb{R}^d$ ($2\leq d\leq 4$). We show the local monotonicity ($p\geq \frac{d}{2}+1,\ r\geq 2$) as well as global monotonicity ($p\geq 2,\ r\geq 4$) 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 (It\^o 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.