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

TL;DR

This paper establishes the existence and uniqueness of solutions for stochastic Ladyzhenskaya-Smagorinsky equations with damping on unbounded domains, and analyzes their small time asymptotics under highly nonlinear unbounded drifts.

Contribution

It proves well-posedness and small time asymptotics for stochastic Ladyzhenskaya-Smagorinsky equations with damping, extending results to unbounded domains and nonlinear drifts.

Findings

Existence of unique strong solutions under certain conditions.

Global and local monotonicity properties of operators.

Small time large deviation principles established.

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.