Journal of the Mathematical Society of Japan

On stability of Leray's stationary solutions of the Navier–Stokes system in exterior domains

Hajime KOBA

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This paper studies the stability of a stationary solution of the Navier–Stokes system in $3$-D exterior domains. The stationary solution is called a Leray's stationary solution if the Dirichlet integral is finite. We apply an energy inequality and maximal $L^p$-in-time regularity for Hilbert space-valued functions to derive the decay rate with respect to time of energy solutions to a perturbed Navier–Stokes system governing a Leray's stationary solution.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 1 (2017), 373-396.

Dates
First available in Project Euclid: 18 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1484730029

Digital Object Identifier
doi:10.2969/jmsj/06910373

Mathematical Reviews number (MathSciNet)
MR3597558

Zentralblatt MATH identifier
1368.35208

Subjects
Primary: 93D20: Asymptotic stability
Secondary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Keywords
asymptotic stability decay property maximal $L^p$ regularity

Citation

KOBA, Hajime. On stability of Leray's stationary solutions of the Navier–Stokes system in exterior domains. J. Math. Soc. Japan 69 (2017), no. 1, 373--396. doi:10.2969/jmsj/06910373. https://projecteuclid.org/euclid.jmsj/1484730029


Export citation

References

  • R. A. Adams and J. J. F. Fournier, Sobolev spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. xiv+305 pp.
  • H-O. Bae and J. Roh, Stability for the 3D Navier–Stokes equations with nonzero far field velocity on exterior domains, J. Math. Fluid Mech., 14 (2012), 117–139.
  • J. Bergh and J. Löfström, Interpolation spaces, An introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976. x+207 pp.
  • W. Borchers and T. Miyakawa, Algebraic $L\sp 2$ decay for Navier–Stokes flows in exterior domains, Acta Math., 165 (1990), 189–227.
  • W. Borchers and T. Miyakawa, On stability of exterior stationary Navier–Stokes flows, Acta Math., 174 (1995), 311–382.
  • L. de Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, (Italian) Rend. Sem. Mat. Univ. Padova, 34 (1964), 205–223.
  • R. Denk, M. Hieber and J. Prüss, $R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114 pp.
  • Y. Enomoto and Y. Shibata, Local energy decay of solutions to the Oseen equation in the exterior domains, Indiana Univ. Math. J., 53 (2004), 1291–1330.
  • Y. Enomoto and Y. Shibata, On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier–Stokes equation, J. Math. Fluid Mech., 7 (2005), 339–367.
  • R. Finn, On steady-state solutions of the Navier–Stokes partial differential equations, Arch. Rational Mech. Anal., 3 (1959), 381–396.
  • G. P. Galdi, An introduction to the mathematical theory of the Navier–Stokes equations, Steady-state problems, Second edition, Springer Monographs in Math., Springer, New York, 2011. xiv+1018 pp.
  • G. P. Galdi, J. G. Heywood and Y. Shibata, On the global existence and convergence to steady state of Navier–Stokes flow past an obstacle that is started from rest, Arch. Rational Mech. Anal., 138 (1997), 307–318.
  • Y. Giga and H. Sohr, Abstract $L\sp p$ estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72–94.
  • H. Heck, H. Kim and H. Kozono, Weak solutions of the stationary Navier–Stokes equations for a viscous incompressible fluid past an obstacle, Math. Ann., 356 (2013), 653–681.
  • J. G. Heywood, On stationary solutions of the Navier–Stokes equations as limits of nonstationary solutions, Arch. Rational Mech. Anal., 37 (1970), 48–60.
  • J. G. Heywood, The exterior nonstationary problem for the Navier–Stokes equations, Acta Math., 129 (1972), 11–34.
  • T. Hishida and Y. Shibata, $L\sb p\text{-}L\sb q$ estimate of the Stokes operator and Navier–Stokes flows in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 193 (2009), 339–421.
  • H. Iwashita, $L^q$-$L^r$ estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier–Stokes initial value problems in $L^q$ spaces, Math. Ann., 285 (1989), 265–288.
  • T. Kato, Strong $L\sp{p}$-solutions of the Navier–Stokes equation in ${\bf R}\sp{m}$, with applications to weak solutions, Math. Z., 187 (1984), 471–480.
  • H. Kim and H. Kozono, On the stationary Navier–Stokes equations in exterior domains, J. Math. Anal. Appl., 395 (2012), 486–495.
  • H. Koba, Nonlinear Stability of Ekman Boundary Layers in Rotating Stratified Fluids, Mem. Amer. Math. Soc., 228 (2014), viii+127 pp.
  • H. Koba, On Stability of the Spatially Inhomogeneous Navier–Stokes–Boussinesq System with General Nonlinearity, Arch. Ration. Mech. Anal., 215 (2015), 907–965.
  • H. Koba, On $L^{3,\infty}$-stability of the Navier–Stokes system in exterior domains, to appear in J. Differential Equations (arXiv:1504.08143).
  • T. Kobayashi and Y. Shibata, On the Oseen equation in the three-dimensional exterior domains, Math. Ann., 310 (1998), 1–45.
  • H. Kozono and T. Ogawa, On stability of Navier–Stokes flows in exterior domains, Arch. Rational Mech. Anal., 128 (1994), 1–31.
  • H. Kozono and M. Yamazaki, Exterior problem for the stationary Navier–Stokes equations in the Lorentz space, Math. Ann., 310 (1998), 279–305.
  • H. Kozono and M. Yamazaki, On a larger class of stable solutions to the Navier–Stokes equations in exterior domains, Math. Z., 228 (1998), 751–785.
  • P. C. Kunstmann and L. Weis, Maximal $L\sb p$-regularity for parabolic equations, Fourier multiplier theorems and $H\sp \infty$-functional calculus, Functional analytic methods for evolution equations, Lecture Notes in Math., 1855, Springer, Berlin, 2004, 65–311.
  • J. Leray, Étude de diverses équations intégrales et de quelques problémes que pose l'Hydrodynamique, J. Math. Pures Appl., 9, (1933), 1–82.
  • A. Lunardi, Interpolation theory, Second edition, Appunti, Scuola Normale Superiore di Pisa (Nuova Serie), [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)] Edizioni della Normale, Pisa, 2009. xiv+191 pp.
  • P. Maremonti, Asymptotic stability theorems for viscous fluid motions in exterior domains, Rend. Sem. Mat. Univ., Padova, 71 (1984), 35–72.
  • K. Masuda, On the stability of incompressible viscous fluid motions past objects, J. Math. Soc. Japan, 27 (1975), 294–327.
  • T. Miyakawa and H. Sohr, On energy inequality, smoothness and large time behavior in $L\sp 2$ for weak solutions of the Navier–Stokes equations in exterior domains, Math. Z., 199 (1988), 455–478.
  • J. Neustupa, Stability of a steady viscous incompressible flow past an obstacle, J. Math. Fluid Mech., 11 (2009), 22–45.
  • A. Novotný and M. Padula, Note on decay of solutions of steady Navier–Stokes equations in $3$-D exterior domains, Differential Integral Equations, 8 (1995), 1833–1842.
  • J. Saal, $\mathscr{R}$-Boundedness, $H^\infty$-calculus, Maximal ($L^p$-)Regularity and Applications to Parabolic PDE's, In Lecture Notes in Math. Sciences, The University of Tokyo, Graduate School of Mathematical Sciences, 2007.
  • Y. Shibata, On an exterior initial-boundary value problem for Navier–Stokes equations, Quart. Appl. Math., 57 (1999), 117–155.
  • M. E. Schonbek, $L\sp 2$ decay for weak solutions of the Navier–Stokes equations, Arch. Rational Mech. Anal., 88 (1985), 209–222.
  • M. E. Schonbek, Large time behaviour of solutions to the Navier–Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733–763.
  • Y. Shibata, On a stability theorem of the Navier–Stokes equation in a three dimensional exterior domain, Tosio Kato's method and principle for evolution equations in mathematical physics (Sapporo, 2001), S$\overline{u}$rikaisekikenky$\overline{u}$sho K$\overline{o}$ky$\overline{u}$roku No. 1234 (2001), 146–172.
  • Y. Shibata and M. Yamazaki, Uniform estimates in the velocity at infinity for stationary solutions to the Navier–Stokes exterior problem, Japan. J. Math. (N.S.), 31 (2005), 225–279.
  • H. Sohr, The Navier–Stokes equations, An elementary functional analytic approach, Birkhäuser Advanced Texts: Basler Lehrbücher, [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2001. x+367 pp.
  • M. Yamazaki, The Navier–Stokes equations in the weak-$L^n$ space with time-dependent external force, Math. Ann., 317 (2000), 635–675.