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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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[1] Bae, H.-O.; Jin, B. J., Upper and lower bounds of temporal and spatial decays for the Navier-Stokes equations, J. Differential Equations, 209, 365-391, (2005) · Zbl 1062.35058
[2] Brandolese, L., On the localization of symmetric and asymmetric solutions of the Navier-Stokes equations in \(\mathbb{R}^n\), C. R. Math. Acad. Sci. Paris, Sér. I, 332, 125-130, (2001) · Zbl 0973.35149
[3] Carpio, A., Large-time behavior in incompressible Navier-Stokes equations, SIAM J. Math. Anal., 27, 449-475, (1996) · Zbl 0845.76019
[4] Coifman, R.; Lions, P. L.; Meyer, Y.; Semmes, S., Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72, 247-286, (1993) · Zbl 0864.42009
[5] Gallay, T.; Wayne, C. E., Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on \(\mathbf{R}^2\), Arch. Ration. Mech. Anal., 163, 209-258, (2002) · Zbl 1042.37058
[6] Gallay, T.; Wayne, C. E., Long-time asymptotics of the Navier-Stokes and vorticity equations on \(\mathbb{R}^3\), Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 360, 2155-2188, (2002) · Zbl 1048.35055
[7] Gallay, T.; Wayne, C. E., Long-time asymptotics of the Navier-Stokes equation in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) [plenary lecture presented at the 76th annual GAMM conference, Luxembourg, 29 March-1 April 2005], ZAMM Z. Angew. Math. Mech., 86, 256-267, (2006)
[8] Fujigaki, Y.; Miyakawa, T., Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the whole space, SIAM J. Math. Anal., 33, 523-544, (2001) · Zbl 0995.35046
[9] Giga, Y.; Matsui, S.; Shimizu, Y., On estimates in Hardy spaces for the Stokes flow in a half space, Math. Z., 231, 383-396, (1999) · Zbl 0928.35118
[10] He, C., Weighted estimates for nonstationary Navier-Stokes equations, J. Differential Equations, 148, 422-444, (1998) · Zbl 0910.35092
[11] He, C.; Xin, Z., On the decay properties of solutions to the non-stationary Navier-Stokes equations in \(\mathbb{R}^3\), Proc. Roy. Soc. Edinburgh Sect. A, 131, 597-619, (2001) · Zbl 0982.35083
[12] He, C.; Miyakawa, T., On two-dimensional Navier-Stokes flows with rotational symmetries, Funkcial. Ekvac., 49, 163-192, (2006) · Zbl 1179.35215
[13] Kajikiya, R.; Miyakawa, T., On \(L^2\) decay of weak solutions of the Navier-Stokes equations in \(\mathbf{R}^n\), Math. Z., 192, 135-148, (1986) · Zbl 0607.35072
[14] T. Kobayashi, T. Kubo, \(L^p\)-\(L^q\) estimates with homogeneous weight for Stokes semigroup and its application to the Navier-Stokes equations, preprint.
[15] Kukavica, I.; Reis, E., Asymptotic expansion for solutions of the Navier-Stokes equations with potential forces, J. Differential Equations, 250, 607-622, (2011) · Zbl 1205.35200
[16] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 193-248, (1934) · JFM 60.0726.05
[17] Miyakawa, T., Application of Hardy space techniques to the time-decay problem for incompressible Navier-Stokes flows in \(\mathbf{R}^n\), Funkcial. Ekvac., 41, 383-434, (1998) · Zbl 1142.35544
[18] Miyakawa, T., On space-time decay properties of nonstationary incompressible Navier-Stokes flows in \(\mathbf{R}^n\), Funkcial. Ekvac., 43, 541-557, (2000) · Zbl 1142.35545
[19] Miyakawa, T., Notes on space-time decay properties of nonstationary incompressible Navier-Stokes flows in \(\mathbf{R}^n\), Funkcial. Ekvac., 45, 271-289, (2002) · Zbl 1141.35434
[20] Miyakawa, T., On upper and lower bounds of rates of decay for nonstationary Navier-Stokes flows in the whole space, Hiroshima Math. J., 32, 431-462, (2002) · Zbl 1048.35063
[21] Miyakawa, T.; Schonbek, M. E., On optimal decay rates for weak solutions to the Navier-Stokes equations in \(\mathbb{R}^n\), Math. Bohem., 126, 2, 443-455, (2001), Proceedings of Partial Differential Equations and Applications (Olomouc, 1999) · Zbl 0981.35048
[22] Okabe, T.; Tsutsui, Y., Navier-Stokes flow in the weighted Hardy space with applications to time decay problem, J. Differential Equations, 261, 1712-1755, (2016) · Zbl 1346.35147
[23] Schonbek, M. E., \(L^2\) decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88, 209-222, (1985) · Zbl 0602.76031
[24] Schonbek, M. E., Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11, 733-763, (1986) · Zbl 0607.35071
[25] Schonbek, M. E., Lower bounds of rates of decay for solutions to the Navier-Stokes equations, J. Amer. Math. Soc., 4, 423-449, (1991) · Zbl 0739.35070
[26] Schonbek, M. E., Asymptotic behavior of solutions to the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 41, 809-823, (1992) · Zbl 0759.35036
[27] Schonbek, M. E.; Schonbek, T. P., On the boundedness and decay of moments of solutions to the Navier-Stokes equations, Adv. Differential Equations, 5, 861-898, (2000) · Zbl 1027.35095
[28] Takahashi, S., A weighted equation approach to decay rate estimates for the Navier-Stokes equations, Nonlinear Anal., 37, 751-789, (1999) · Zbl 0941.35066
[29] Tsutsui, Y., An application of weighted Hardy spaces to the Navier-Stokes equations, J. Funct. Anal., 266, 1395-1420, (2014) · Zbl 1306.46040
[30] Wiegner, M., Decay results for weak solutions of the Navier-Stokes equations on \(\mathbf{R}^n\), J. Lond. Math. Soc. (2), 35, 303-313, (1987) · Zbl 0652.35095
[31] Wiegner, M., Decay of the \(L_\infty\)-norm of solutions of Navier-Stokes equations in unbounded domains, Acta Appl. Math., 37, 215-219, (1994) · Zbl 0816.35107
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