an:06807482
Zbl 1378.35225
Okabe, Takahiro
Space-time asymptotics of the two dimensional Navier-Stokes flow in the whole plane
EN
J. Differ. Equations 264, No. 2, 728-754 (2018).
00371793
2018
j
35Q30 76D05 35C20 35B40 30H10
time decay; asymptotic expansion; large time behavior
Summary: We consider the space-time behavior of the two dimensional Navier-Stokes flow. Introducing some qualitative structure of initial data, we succeed to derive the first order asymptotic expansion of the Navier-Stokes flow without moment condition on initial data in \(L^1(\mathbb{R}^2) \cap L_{\sigma}^2(\mathbb{R}^2)\). Moreover, we characterize the necessary and sufficient condition for the rapid energy decay \(||u(t)||_2=o(t^{-1})\) as \(t \to \infty\) motivated by \textit{T. Miyakawa} and \textit{M. E. Schonbek} [Math. Bohem. 126, No. 2, 443--455 (2001; Zbl 0981.35048)]. By weighted estimated in Hardy spaces, we discuss the possibility of the second order asymptotic expansion of the Navier-Stokes flow assuming the first order moment condition on initial data. Moreover, observing that the Navier-Stokes flow \(u(t)\) lies in the Hardy space \(H^1(\mathbb{R}^2)\) for \(t>0\), we consider the asymptotic expansions in terms of Hardy-norm. Finally we consider the rapid time decay \(||u(t)||_2=o(t^{-\frac{3}{2}})\) as \(t \to \infty\) with cyclic symmetry introduced by \textit{L. Brandolese} [C. R. Acad. Sci., Paris, S??r. I, Math. 332, No. 2, 125--130 (2001; Zbl 0973.35149)].
Zbl 0981.35048; Zbl 0973.35149