Exponential Decay and Regularity of Global Solutions for the 3D Navier–Stokes Equations Posed on Lipschitz and Smooth Domains

Abstract

We study initial-boundary value problems for the 3D Navier–Stokes equations posed on bounded and unbounded parallelepipeds as well as on bounded and unbounded smooth domains without smallness restrictions for the initial data. Under conditions on sizes of domains, we establish the existence, uniqueness and exponential decay of solutions in $$H^2$$ -norm for bounded domains as well as “smoothing” effect and in $$H^1$$ -norm for unbounded ones. Moreover, for smooth subdomains of unbounded domains, we prove regularity of strong solutions and “smoothing” effect.