Decay properties of strong solutions for the Navier-Stokes equations in two-dimensional unbounded domains
The initial-boundary value problem for the Navier-Stokes system is considered in an unbounded domain \(\Omega\subset \mathbb{R}^ 2\), \[ \begin{cases} u_ t- \Delta u+ u\cdot \nabla u+\nabla p=0,\quad\text{div }u=0,\quad x\in \Omega,\;t>0,\\ u=0\quad x\in\partial \Omega,\;t>0;\quad u|_{t=0}=a.\end{cases}\tag{1} \] Let \(\|\cdot\|_ q\) be the \(L^ q\)-norm on \(\Omega\) and \(A\) the Stokes operator in \(L^ 2\). The main result of the paper is the description of the behaviour of a strong solution of the problem (1) as \(t\to\infty\). If \(0\leq \mu<{1\over 2}\) and \(a\in\mathbb{R}(A^ \mu)\) then the strong solution \(u\) of (1) has the following decay properties \[ \| A^ \alpha u(t)\|_ 2=o(t^{- \mu-\alpha}),\;\| Au(t)\|_ 2=o(t^{-\mu-1}\sqrt{\log t}),\;\Bigl\| A^ \alpha {\partial\over\partial t} u(t)\Bigr\|_ 2= o(t^{-\mu-\alpha-1}) \] for \(0\leq \alpha<1\). Analogous estimates are obtained for \(L^ q\)-norms of the solution for \(q\geq 2\). New proofs for some results of \textit{W. Borchers} and \textit{T. Miyakawa} [Arch. Ration. Mech. Anal. 118, No. 3, 273-295 (1992; Zbl 0756.76018)] are adduced in the paper.
scientific article; zbMATH DE number 399275
scientific article