On uniform stabilization of solutions of the exterior problem for the Navier–Stokes equations

The first mixed problem with homogeneous boundary conditions for the system of Stokes and Navier–Stokes equations is considered in a cylinder $D=(0,\infty)\times\Omega$, where $\Omega$ is the complement of the closure of a bounded domain in $R^3$. For solutions of both problems uniform decay with rate $t^{-3/2}$ is proved under certain smoothness conditions on the boundary under the assumption that the initial vector belongs to $\mathbf{L}_2$. Here in the case of the nonlinear problem it is additionally assumed that a weak solution satisfies the strong energy inequality.
A result on the decay of a solution of the linearized system of Navier–Stokes equations is used in the proof of the main assertion on stabilization of a solution of the problem with a bounded initial vector-valued function: existence of a uniform zero spherical limit mean of the initial function is necessary and sufficient for uniform stabilization of the solution to zero.