Large deviations of fully local monotone stochastic partial differential equations driven by gradient-dependent noise

TL;DR

This paper establishes a large deviation principle for solutions of a broad class of fully local monotone SPDEs with gradient-dependent noise, extending existing results to more general models.

Contribution

It introduces a new framework for LDP in fully local monotone SPDEs, allowing the diffusion coefficient to depend on the solution's gradient, covering many complex models.

Findings

Proves a small noise large deviation principle for a wide class of SPDEs.

Includes models like stochastic Navier-Stokes and p-Laplace equations with gradient-dependent noise.

Uses pseudomonotone and compactness techniques for the analysis.

Abstract

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H\subseteq V^*$ $$ \left\{ \begin{align} &dX_t=A(t,X_t)dt+B(t,X_t)dW_t,\ t\in (0,T]\\\\& X_0=x\in H, \end{align} \right. $$ where $$A: [0,T] \times V\rightarrow V^*,\ \ B:[0,T]\times V\rightarrow\ L_2(U,H)$$ are measurable maps, $L_2(U,H)$ is the space of Hilbert-Schmidt operators from $U$ to $H$ and $W$ is a $U$-cylindrical Wiener process.\par In this paper, we establish a small noise large deviation principle(LDP) for the solutions {$u^\varepsilon$}$_{\varepsilon>0}$ of the above SPDEs. The main contribution of this paper is the much more generality of our framework than that of the existing results. In particular, the diffusion coefficient $B(t,\cdot)$ may depend on the gradient of the solutions, which is of great interest in the field of SPDEs, but there are few existing results on the topic of LDP. The broader scope of the fully local monotone setting leads us to use different strategies and techniques. A combination of the pseudomonotone technique and compactness arguement plays a crucial role in the whole paper. Our framework is very general to include many interesting models that could not be covered by existing work, including stochastic quasilinear SPDEs, stochastic convection diffusion equation, stochastic 2D Liquid crystal equation, stochastic $p$-Laplace equation with gradient-dependent noise, stochastic 2D Navier-Stokes equation with gradient-dependent noise etc.