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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
##### Keywords:
time decay; asymptotic expansion; large time behavior
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##### References:
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