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

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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.