Asymptotic log-Harnack inequality for the 3D stochastic globally modified Allen-Cahn-Navier-Stokes system with degenerate noise

Abstract

In this article, we consider a globally modified Allen-Cahn-Navier-Stokes equations in a three dimensional bounded domain and examine some asymptotic behaviors of the strong solution. More precisely, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with the globally modified Allen-Cahn-Navier-Stokes system driven by an additive degenerate noise via the asymptotic coupling method. As consequences of the asymptotic log-Harnack inequality, we derive the gradient estimate, the asymptotic irreducibility, the asymptotic strong Feller property, the asymptotic heat kernel estimate and the ergodicity.