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Pointwise Decay in Space and in Time for Incompressible Viscous Flow Around a Rigid Body Moving with Constant Velocity

  • Paul DeuringEmail author
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Abstract

We present pointwise space–time decay estimates for the velocity part of solutions to the time-dependent Oseen system in 3D, with Dirichlet boundary conditions and vanishing velocity at infinity. In addition, similar estimates are derived for solutions to the time-dependent incompressible Navier–Stokes system with Oseen term, and for solutions to the stability problem associated with the stationary incompressible Navier–Stokes system with Oseen term.

Keywords

Incompressible Navier–Stokes system Oseen term Decay 

Mathematics Subject Classification

35Q30 65N30 76D05 
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Notes

Compliance with Ethical Standards

Conflict of interest

The author declares that he has no conflict of interest.

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Babenko, K.I., Vasil’ev, M.M.: On the asymptotic behavior of a steady flow of viscous fluid at some distance from an immersed body. Prikl. Mat. Meh. 37, : 690–705 (Russian); English translation. J. Appl. Math. Mech. 37(1973), 651–665 (1973)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bae, H.-O., Jin, B.J.: Estimates of the wake for the 3D Oseen equations. Discret. Contin. Dyn. Syst. Ser. B 10, 1–18 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bae, H.-O., Roh, J.: Stability for the 3D Navier–Stokes equations with nonzero far field velocity on exterior domains. J. Math. Fluid Mech. 14, 117–139 (2012)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Deuring, P.: Exterior stationary Navier–Stokes flows in 3D with nonzero velocity at infinity: asymptotic behaviour of the velocity and its gradient. IASME Trans. 6, 900–904 (2005)MathSciNetGoogle Scholar
  6. 6.
    Deuring, P.: The single-layer potential associated with the time-dependent Oseen system. In: Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics. pp. 117–125. Chalkida, Greeece, (2006)Google Scholar
  7. 7.
    Deuring, P.: On volume potentials related to the time-dependent Oseen system. WSEAS Trans. Math. 5, 252–259 (2006)MathSciNetGoogle Scholar
  8. 8.
    Deuring, P.: On boundary driven time-dependent Oseen flows. Banach Center Publ. 81, 119–132 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Deuring, P.: A potential theoretic approach to the time-dependent Oseen system. In: Rannacher, R., Sequeira, A. (eds.) Advances in Mathematical Fluid Mechanics. Dedicated to Giovanni Paolo Galdi on the Occasion of his 60th Birthday. pp. 191–214. Springer, Berlin, (2010)Google Scholar
  10. 10.
    Deuring, P.: Spatial decay of time-dependent Oseen flows. SIAM J. Math. Anal. 41, 886–922 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Deuring, P.: A representation formula for the velocity part of 3D time-dependent Oseen flows. J. Math. Fluid Mech. 16, 1–39 (2014)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Deuring, P.: The Cauchy problem for the homogeneous time-dependent Oseen system in \( {\mathbb{R}}^3 \): spatial decay of the velocity. Math. Bohemica 138, 299–324 (2013)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Deuring, P.: Pointwise spatial decay of time-dependent Oseen flows: the case of data with noncompact support. Discrete Contin. Dyn. Syst. Ser. A 33, 2757–2776 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Deuring, P.: Spatial decay of time-dependent incompressible Navier–Stokes flows with nonzero velocity at infinity. SIAM J. Math. Anal. 45, 1388–1421 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Deuring, P., Kračmar, S.: Exterior stationary Navier–Stokes flows in 3D with non-zero velocity at infinity: approximation by flows in bounded domains. Math. Nachr. 269–270, 86–115 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Enomoto, Y., Shibata, Y.: Local energy decay of solutions to the Oseen equation in the exterior domain. Indiana Univ. Math. J. 53, 1291–1330 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Enomoto, Y., Shibata, Y.: On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier–Stokes equation. J. Math. Fluid Mech. 7, 339–367 (2005)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Farwig, R.: The stationary exterior 3D-problem of Oseen and Navier–Stokes equations in anisotropically weighted Sobolev spaces. Math. Z. 211, 409–447 (1992)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Fučik, S., John, O., Kufner, A.: Function Spaces. Noordhoff, Leyden (1977)zbMATHGoogle Scholar
  20. 20.
    Galdi, g P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, 2nd edn. Springer, New York (2011)zbMATHGoogle Scholar
  21. 21.
    Heywood, J.G.: The exterior nonstationary problem for the Navier–Stokes equations. Acta Math. 129, 11–34 (1972)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Heywood, J.G.: The Navier–Stokes equations. On the existence, regularity and decay of solutions. Indiana Univ. Math. J 29, 639–681 (1980)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Knightly, G.H.: Some decay properties of solutions of the Navier–Stokes equations. In: Rautmann, R. (ed.): Approximation methods for Navier–Stokes problems. Lecture Notes in Mathematics, vol. 771, pp. 287–298. Springer, Berlin, (1979)Google Scholar
  24. 24.
    Kobayashi, T., Shibata, Y.: On the Oseen equation in three-dimensional exterior domains. Math. Ann. 310, 1–45 (1998)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Masuda, K.: On the stability of incompressible viscous fluid motions past bodies. J. Math. Soc. Jpn. 27, 294–327 (1975)CrossRefGoogle Scholar
  26. 26.
    Miyakawa, T.: On nonstationary solutions of the Navier–Stokes equations in an exterior domain. Hiroshima Math. J. 12, 115–140 (1982)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mizumachi, R.: On the asymptotic behaviour of incompressible viscous fluid motions past bodies. J. Math. Soc. Jpn. 36, 497–522 (1984)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Neustupa, J.: Stability of a steady viscous incompressible flow past an obstacle. J. Math. Fluid Mech. 11, 22–45 (2009)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Neustupa, J.: A spectral criterion for stability of a steady viscous incompressible flow past an obstacle. J. Math. Fluid Mech. 18, 133–156 (2016)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Shen, Zongwei: Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders. Am. J. Math. 113, 293–373 (1991)CrossRefGoogle Scholar
  31. 31.
    Shibata, Y.: On an exterior initial boundary value problem for Navier-Stokes equations. Q. Appl. Math. 57, 117–155 (1999)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Solonnikov, V.A.: A priori estimates for second order parabolic equations. Trudy Mat. Inst. Steklov. 70: 133–212 (Russian). English translation. AMS Trans. 65(1967), 51–137 (1964)Google Scholar
  33. 33.
    Solonnikov, V.A.: Estimates for solutions of nonstationary Navier–Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38: 153–231 (Russian); English translation. J. Soviet Math. 8(1977), 467–529 (1973)Google Scholar
  34. 34.
    Stein, E.M.: Singular Integrals and Differentiability of Functions. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Pures et Appliquées Joseph LiouvilleUniversité du Littoral Côte d’OpaleCalaisFrance

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