## On the spatially asymptotic structure of time-periodic solutions to the Navier–Stokes equations

HTML articles powered by AMS MathViewer

- by Thomas Eiter PDF
- Proc. Amer. Math. Soc.
**149**(2021), 3439-3451 Request permission

## Abstract:

The asymptotic behavior of weak time-periodic solutions to the Navier–Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions.## References

- K. I. Babenko,
*The stationary solutions of the problem of the flow around a body by a viscous incompressible fluid*, Mat. Sb. (N.S.)**91(133)**(1973), 3–26, 143 (Russian). MR**0348301** - François Bruhat,
*Distributions sur un groupe localement compact et applications à l’étude des représentations des groupes $\wp$-adiques*, Bull. Soc. Math. France**89**(1961), 43–75 (French). MR**140941**, DOI 10.24033/bsmf.1559 - T. Eiter,
*Existence and Spatial Decay of Periodic Navier-Stokes Flows in Exterior Domains*, Logos Verlag Berlin, 2020. - T. Eiter and G. P. Galdi,
*Spatial decay of the vorticity field of time-periodic viscous flow past a body*, arXiv:2011.12579, 2020. - T. Eiter and M. Kyed,
*Time-periodic linearized Navier-Stokes equations: an approach based on Fourier multipliers*, Particles in flows, Adv. Math. Fluid Mech., Birkhäuser/Springer, Cham, 2017, pp. 77–137. MR**3727766** - Thomas Eiter and Mads Kyed,
*Estimates of time-periodic fundamental solutions to the linearized Navier-Stokes equations*, J. Math. Fluid Mech.**20**(2018), no. 2, 517–529. MR**3808582**, DOI 10.1007/s00021-017-0332-7 - R. Farwig,
*Das stationäre Außenraumproblem der Navier-Stokes-Gleichungen bei nichtverschwindender Anströmgeschwindigkeit in anisotrop gewichteten Sobolevräumen*, SFB 256 preprint no. 110 (Habilitationsschrift). University of Bonn (1990). - Reinhard Farwig,
*The stationary exterior $3$D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces*, Math. Z.**211**(1992), no. 3, 409–447. MR**1190220**, DOI 10.1007/BF02571437 - Robert Finn,
*An energy theorem for viscous fluid motions*, Arch. Rational Mech. Anal.**6**(1960), 371–381. MR**166497**, DOI 10.1007/BF00276169 - Robert Finn,
*Estimates at infinity for stationary solutions of the Navier-Stokes equations*, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.)**3(51)**(1959), 387–418. MR**166495** - Robert Finn,
*On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems*, Arch. Rational Mech. Anal.**19**(1965), 363–406. MR**182816**, DOI 10.1007/BF00253485 - Giovanni Galdi and Hermann Sohr,
*Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flow past a body*, Arch. Ration. Mech. Anal.**172**(2004), no. 3, 363–406. MR**2062429**, DOI 10.1007/s00205-004-0306-9 - G. P. Galdi,
*An introduction to the mathematical theory of the Navier-Stokes equations*, 2nd ed., Springer Monographs in Mathematics, Springer, New York, 2011. Steady-state problems. MR**2808162**, DOI 10.1007/978-0-387-09620-9 - Giovanni P. Galdi and Mads Kyed,
*Time-periodic solutions to the Navier-Stokes equations in the three-dimensional whole-space with a non-zero drift term: asymptotic profile at spatial infinity*, Mathematical analysis in fluid mechanics—selected recent results, Contemp. Math., vol. 710, Amer. Math. Soc., [Providence], RI, [2018] ©2018, pp. 121–144. MR**3818671**, DOI 10.1090/conm/710/14367 - Loukas Grafakos,
*Classical Fourier analysis*, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR**2445437** - Stanislav Kračmar, Antonín Novotný, and Milan Pokorný,
*Estimates of Oseen kernels in weighted $L^p$ spaces*, J. Math. Soc. Japan**53**(2001), no. 1, 59–111. MR**1800524**, DOI 10.2969/jmsj/05310059 - M. Kyed,
*Time-Periodic Solutions to the Navier-Stokes Equations*, Habilitationsschrift, Technische Universität Darmstadt, 2012.

## Additional Information

**Thomas Eiter**- Affiliation: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany
- Email: thomas.eiter@wias-berlin.de
- Received by editor(s): May 27, 2020
- Received by editor(s) in revised form: December 8, 2020
- Published electronically: May 12, 2021
- Communicated by: Catherine Sulem
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 3439-3451 - MSC (2020): Primary 35Q30, 35B10, 35C20, 76D05, 35E05
- DOI: https://doi.org/10.1090/proc/15482
- MathSciNet review: 4273147