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

F. Kh. Mukminov

Sbornik. Mathematics, Tome 81 (1995) no. 2, pp. 297-320

Résumé

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.