A viscosity solution approach to the large deviation principle for stochastic convective Brinkman-Forchheimer equations

Abstract

This article develops the viscosity solution approach to the large deviation principle for the following two- and three-dimensional stochastic convective Brinkman-Forchheimer equations on the torus $\mathbb{T}^d,\ d\in\{2,3\}$ with small noise intensity: \begin{align*} \mathrm{d}\boldsymbol{u}_n+[-\mu\Delta\boldsymbol{u}_n+ (\boldsymbol{u}_n\cdot\nabla)\boldsymbol{u}_n +\alpha\boldsymbol{u}_n+\beta|\boldsymbol{u}_n|^{r-1}\boldsymbol{u}_n+\nabla p_n]\mathrm{d} t=\boldsymbol{f}\mathrm{d} t+\frac{1}{\sqrt{n}}\mathrm{Q}^{\frac12}\mathrm{d}\mathrm{W}, \ \nabla\cdot\boldsymbol{u}_n=0, \end{align*} where $\mu,\alpha,\beta>0$, $r\in[1,\infty)$, $\mathrm{Q}$ is a trace class operator and $\mathrm{W}$ is Hilbert-valued calendrical Wiener process. We build our analysis on the framework of Varadhan and Bryc, together with the techniques of [J. Feng et.al., Large Deviations for Stochastic Processes, American Mathematical Society (2006) vol. \textbf{131}]. By employing the techniques from the comparison principle, we identify the Laplace limit as the convergence of the viscosity solution of the associated second-order singularly perturbed Hamilton-Jacobi-Bellman equation. A key advantage of this method is that it establishes a Laplace principle without relying on additional sufficient conditions such as Bryc's theorem, which the literature commonly requires. For $r>3$ and $r=3$ with $2\beta\mu\geq1$, we also derive the exponential moment bounds without imposing the classical orthogonality condition $((\boldsymbol{u}_n\cdot\nabla)\boldsymbol{u}_n,\mathrm{A}\boldsymbol{u}_n)=0$, where $\mathrm{A}=-\Delta$, in both two-and three-dimensions. We first establish the large deviation principle in the Skorohod space. Then, by using the $\mathrm{C}-$exponential tightness, we finally establish the large deviation principle in the continuous space.