Stabilization of the solution of a two-dimensional system of Navier–Stokes equations in an unbounded domain with several exits to infinity

The behaviour as $t\to\infty$ of the solution of the mixed problem for the system of Navier–Stokes equations with a Dirichlet condition at the boundary is studied in an unbounded two-dimensional domain with several exits to infinity. A class of domains is distinguished in which an estimate characterizing the decay of solutions in terms of the geometry of the domain is proved for exponentially decreasing initial velocities. A similar estimate of the solution of the first mixed problem for the heat equation is sharp in a broad class of domains with several exits to infinity.