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On the decay of higher-order norms of the solutions of Navier–Stokes equations

Published online by Cambridge University Press:  14 November 2011

Maria E. Schonbek  and
Michael Wiegner
Maria E. Schonbek
Affiliation:
Department of Mathematics, University of California, Santa Cruz, California 95064, U.S.A.
Michael Wiegner
Affiliation:
Department of Mathematics, University of Bayreuth, D-95440 Bayreuth, Germany
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Extract

We show that an energy decay ∥u(t)∥2 = O(t−µ) for solutions of the Navier–Stokes equations on ℝn, n ≦ 5, implies a decay of the higher order norms, e.g. ∥Dα u(t)∥2 = O(t−µ −|α|/2) and ∥u(t)|∞ = O(t−µ −n/4).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 3 , 1996 , pp. 677 - 685
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Borchers, W. and Miyakawa, T.. L2 decay for the Navier–Stokes flows in unbounded domains with application to exterior stationary flows. Arch. Rational Mech. Anal. 118 (1992), 273–95.CrossRefGoogle Scholar
2Borchers, W.. and Varnhorn, W.. On the boundedness of the Stokes semigroup in two-dimensional exterior domains. Math. Z. 213 (1993), 275–99.CrossRefGoogle Scholar
3Galdi, G. P. and Maremonti, P.. Monotonic decreasing and asymptotic behaviour of the kinetic energy for weak solutions of the Navier–Stokes equations in exterior domains. Arch. Rational Mech. Anal. 94(1986), 253–66.CrossRefGoogle Scholar
4Heywood, J. G.. The Navier–Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29 (1980), 639–81.CrossRefGoogle Scholar
5Iwashita, H.. Lq – Lr, estimates for solutions of nonstationary Stokes equations in an exterior domain and the Navier–Stokes initial value problem in Lq spaces. Math. Ann. 285 (1989), 265–88.CrossRefGoogle Scholar
6Kagei, Y. and Wahl, W. von. Stability of higher norms in terms of energy-stability for the Boussinesqequations. Remarks on the asymptotic behaviour of convection-roll-type solutions. Differential and Integral Equations 1 (1994), 921–48.Google Scholar
7Kajikiya, R. and Miyakawa, T.. On L2 decay of weak solutions of the Navier–Stokes equations in ℝn. Math. Z. 192 (1986), 135–48.CrossRefGoogle Scholar
8Kato, T.. Strong Lp-solutions of the Navier–Stokes equation in ℝm, with applications to weak solutions. Math. Z. 187 (1984), 471–80.CrossRefGoogle Scholar
9Kozono, H., Ogawa, T. and Sohr, H.. Asymptotic behaviour in Lr for turbulent solutions of the Navier–Stokes equations in exterior domains. Manusripta Math. 74 (1992), 253–75.CrossRefGoogle Scholar
10Kozono, H. and Ogawa, T.. Decay properties of strong solutions for the Navier–Stokes equations in two-dimensional unbounded domains. Arch. Rational Mech. Anal. 122 (1993), 1–17.CrossRefGoogle Scholar
11Maremonti, P.. On the asymptotic behaviour of the L2-norm of suitable weak solutions to the Navier–Stokes equations in three-dimensional exterior domains. Comm. Math. Phys. 118 (1988), 385–400.CrossRefGoogle Scholar
12Miyakawa, T. and Sohr, H.. On energy inequality, smoothness and large time behaviour in L2 for weak solutions of the Navier–Stokes equations in exterior domains. Math. Z. 199 (1988), 455–78.CrossRefGoogle Scholar
13Schonbek, M. E.. L2-decay for weak solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal 88(1985), 209–22.CrossRefGoogle Scholar
14Schonbek, M. E.. Large-time behaviour of solutions of the Navier–Stokes equations. Comm. Partial Differential Equations 11 (1986), 753–63.CrossRefGoogle Scholar
15Schonbek, M. E.. Lower bounds of rates of decay for solutions to the Navier–Stokes equations. J. Amer. Math. Soc. 4 (1991), 423–49.CrossRefGoogle Scholar
16Schonbek, M. E.. Large time behaviour of solutions to the Navier–Stokes equations in Hm spaces (Preprint, 1993).CrossRefGoogle Scholar
17Sohr, H., Wahl, W. von and Wiegner, M.. Zur Asymptotik der Gleichungen von Navier–Stokes. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 3 (1986), 49–59.Google Scholar
18Wiegner, M.. Decay results for weak solutions of the Navier–Stokes equations in ℝn. J. London Math. Soc. 35(1987), 303–13.CrossRefGoogle Scholar
19Wiegner, M.. Decay and stability in Lp for strong solutions of the Cauchy problem for the Navier–Stokes equations. In The Navier–Stokes Equations, Theory and Numerical Methods, eds Heywood, J. G. et al. , Proceedings Oberwolfach, Lecture Notes in Mathematics 1431, 95–9 (Berlin: Springer, 1990).Google Scholar
20Wiegner, M.. Decay of the L∞-norm of solutions of the Navier–Stokes equations in unbounded domains. In Proceedings of ‘Mathematical problems for Navier–Stokes equations, Cento 1993’, Acta Appl. Math. 37 (1994), 215–19.Google Scholar