Uniform Large Deviation Principle for the Solutions of Two-Dimensional Stochastic Navier–Stokes Equations in Vorticity Form

Abstract

The main objective of this paper is to demonstrate the uniform large deviation principle (UDLP) for the solutions of two-dimensional stochastic Navier–Stokes equations (SNSE) in the vorticity form when perturbed by two distinct types of noises. We first consider an infinite-dimensional additive noise that is white in time and colored in space and then consider a finite-dimensional Wiener process with linear growth coefficient. In order to obtain the ULDP for 2D SNSE in the vorticity form, where the noise is white in time and colored in space, we utilize the existence and uniqueness result from B. Ferrario et. al., Stochastic Process. Appl., 129 (2019), 1568–1604, and the uniform contraction principle. For the finite-dimensional multiplicative Wiener noise, we first prove the existence of a unique local mild solution to the vorticity equation using a truncation and fixed point arguments. We then establish the global existence of the truncated system by deriving a uniform energy estimate for the local mild solution. By applying stopping time arguments and a version of Skorokhod’s representation theorem, we conclude the global existence and uniqueness of a solution to our model. We employ the weak convergence approach to establish the ULDP for the law of the solutions in two distinct topologies. We prove ULDP in the \({{\textrm{C}}([0,T];{\textrm{L}}^p({\mathbb {T}}^2))}\) topology, for \(p>2\), taking into account the uniformity of the initial conditions contained in bounded subsets of \({{\textrm{L}}^p({\mathbb {T}}^2)}\). Finally, in \({{\textrm{C}}([0,T]\times {\mathbb {T}}^2)}\) topology, the uniformity of initial conditions lying in bounded subsets of \({{\textrm{C}}({\mathbb {T}}^2)}\) is considered.