Global strong solutions of Navier–Stokes equations with interface boundary in three-dimensional thin domains

In this article, we study the spectrum of the Stokes operator in a 3D two layer domain with interface, obtain the asymptotic estimates on the spectrum of the Stokes operator as thickness εε goes to zero. Based on the spectral decomposition of the Stokes operator, a new average-like operator is introduced and applied to the study of Navier–Stokes equation in the two layer thin domains under interface boundary condition. We prove the global existence of strong solutions to the 3D Navier–Stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small. This article is a continuation of our study on the Stokes operator under Navier friction boundary condition. Due to the viscosity distinction between the two layers, the Stokes operator displays radically different spectral structure from that under Navier friction boundary condition, then causes great difficulty to the analysis.