Large deviation principles for 2D stochastic $g$-Navier--Stokes equations on unbounded channel-like domains

Abstract

This paper investigates the large deviation principle for 2D stochastic $g$-Navier--Stokes equations driven by multiplicative noise and defined on an unbounded channel-like domain. We first prove the existence and uniqueness of probabilistically strong solutions for the stochastic equation under small noise perturbations by means of the local monotonicity method. Subsequently, we show the well-posedness of solutions to the corresponding control equation and the convergence in distribution of the solutions for the stochastic equation as the noise intensity tends to zero. Finally, we establish the Freidlin-Wentzell type large deviation principle for the solutions of the stochastic equation by using the weak convergence method. To overcome the lack of compactness of Sobolev embedding on unbounded domains, we employ a scalar stream function together with a truncation technique to derive uniform tail estimates for the solutions of the control equation.