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Space-time asymptotics of the two dimensional Navier-Stokes flow in the whole plane. (English) Zbl 1378.35225

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 T. Miyakawa and 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 L. Brandolese [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 2, 125–130 (2001; Zbl 0973.35149)].

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35C20 Asymptotic expansions of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
30H10 Hardy spaces
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