Abstract
We investigate the pointwise behavior of time-periodic Navier–Stokes flows in the whole space. We show that if the time-periodic external force is sufficiently small in an appropriate sense, then there exists a unique time-periodic solution \(\{ u,p \}\) of the Navier–Stokes equation such that \(|u(t,x)|=O(|x|^{1-n})\), \(|\nabla u(t,x)|=O(|x|^{-n})\) and \(|p(t,x)|=O(|x|^{-n})\) uniformly in \(t \in {\mathbb {R}}\) as \(|x| \rightarrow \infty\). Our solution decays more rapidly than the time-periodic Stokes fundamental solution. The proof is based on the representation formula of a solution via the time-periodic Stokes fundamental solution and its properties.
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References
Eiter, T., Kyed, M.: Estimates of time-periodic fundamental solutions to the linearized Navier–Stokes equations. J. Math. Fluid Mech. (to appear)
Farwig, R., Nakatsuka, T., Taniuchi, Y.: Uniqueness of solutions on the whole time axis to the Navier–Stokes equations in unbounded domains. Commun. Partial Differ. Equ. 40, 1884–1904 (2015)
Galdi, G.P., Sohr, H.: Existence and uniqueness of time-periodic physically reasonable Navier–Stokes flow past a body. Arch. Ration. Mech. Anal. 172, 363–406 (2004)
Kang, K., Miura, H., Tsai, T.-P.: Asymptotics of small exterior Navier–Stokes flows with non-decaying boundary data. Commun. Partial Differ. Equ. 37, 1717–1753 (2012)
Kozono, H., Nakao, M.: Periodic solutions of the Navier–Stokes equations in unbounded domains. Tohoku Math. J. 48, 33–50 (1996)
Kozono, H., Sohr, H.: New a priori estimates for the Stokes equations in exterior domains. Indiana Univ. Math. J. 40, 1–27 (1991)
Kyed, M.: Time-periodic solutions to the Navier–Stokes equations. Habilitationsschrift, Technische Universität Darmstadt (2012)
Kyed, M.: Maximal regularity of the time-periodic linearized Navier–Stokes system. J. Math. Fluid Mech. 16, 523–538 (2014)
Kyed, M.: A fundamental solution to the time-periodic Stokes equations. J. Math. Anal. Appl. 437, 708–719 (2016)
Maremonti, P.: Existence and stability of time-periodic solutions to the Navier–Stokes equations in the whole space. Nonlinearity 4, 503–529 (1991)
Miyakawa, T.: On Stationary Incompressible Navier–Stokes Flows with Fast Decay and the Vanishing Flux Condition, Topics in Nonlinear Analysis, Prog. Nonlinear Differ. Equ. Appl. vol. 35, pp 535–552. Birkhäuser, Basel (1999)
Nakatsuka, T.: Note on the fast decay property of steady Navier–Stokes flows in the whole space. Preprint Math. Inst. CAS No. IM-2017-15
Silvestre, A.L.: Existence and uniqueness of time-periodic solutions with finite kinetic energy for the Navier–Stokes equations in \(\mathbb{R}^n\). Nonlinearity 25, 37–55 (2012)
Taniuchi, Y.: On the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domains. Math. Z. 261, 597–615 (2009)
Yamazaki, M.: The Navier–Stokes equations in the weak-\(L^n\) space with time-dependent external force. Math. Ann. 317, 635–675 (2000)
Acknowledgements
The research was supported by the Academy of Sciences of the Czech Republic, Institute of Mathematics (RVO: 67985840). The author would like to thank Professor M. Kyed for useful comments.
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Nakatsuka, T. On time-periodic Navier–Stokes flows with fast spatial decay in the whole space. J Elliptic Parabol Equ 4, 51–67 (2018). https://doi.org/10.1007/s41808-018-0011-8
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DOI: https://doi.org/10.1007/s41808-018-0011-8