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Pointwise Spatial Decay of Weak Solutions to the Navier–Stokes System in 3D Exterior Domains

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Abstract

We consider L 2-weak solutions to the Navier–Stokes system in a 3D exterior domain. Under the assumptions that the initial data decrease as \({O( |x|^{-\mu})}\) when \({|x| \to \infty}\), for some \({\mu \geq 7/6}\), and that the volume force decays sufficiently fast, we show that the velocity decreases pointwise with the rate \({O(\, |x|^{-\min\{\mu ,13/5\}})}\) for \({|x| \to \infty}\), uniformly with respect to time.

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References

  1. Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Adams R.A., Fournier J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, Amsterdam (2003)

    MATH  Google Scholar 

  3. Amrouche C., Girault V., Schonbek M.E., Schonbek T.P.: Pointwise decay of solutions and of higher derivatives to Navier–Stokes equations. SIAM J. Math. Anal. 31, 740–753 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Babenko, K.I.: On stationary solutions of the problem of flow past a body of a viscous incompressible fluid. Math. Sbornik 91, 3–26 (1973) (Russian) [English translation, Math. USSR Sbornik 20, 1–25 (1973)]

  5. Bae H.-O., Brandolese L.: On the effect of external forces on incompressible fluid motions at large distances. Ann. Univ. Ferrara 55, 225–238 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bae H.-O., Hi J.C.: Decay rate for incompressible flows in half space. Math. Z. 238, 799–816 (2001)

    MATH  MathSciNet  Google Scholar 

  7. Bae H.-O., Bum J.J.: Upper and lower bounds of temporal and spatial decays for the Navier–Stokes equations. J. Differ. Equ. 209, 365–391 (2005)

    Article  ADS  MATH  Google Scholar 

  8. Bae H.-O., Bum J.J.: Temporal and spatial decays for the Navier–Stokes equations. Proc. R. Soc. Edinb. Sect. A 135, 461–477 (2005)

    Article  MATH  Google Scholar 

  9. Bae H.-O., Roh J.: Weighted estimates for the incompressible fluid in exterior domains. J. Math. Anal. Appl. 355, 846–854 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bae H.-O., Roh J.: Optimal weighted estimates of flows in exterior domains. Nonlinear Anal. 73, 1350–1363 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Borchers W., Sohr H.: On the equations rot v = g and div u = f with zero boundary conditions. Hokkaido Math. J. 19, 67–87 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Borchers W., Sohr H.: On the semigroup of the Stokes operator for exterior domains in L q-spaces. Math. Z. 196, 415–425 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  13. Brandolese L.: Asymptotic behavior of the energy and pointwise estimates for solutions to the Navier–Stokes equation. Rev. Mat. Iberoamericana 20, 223–256 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Brandolese L.: Space-time decay of Navier–Stokes flows invariant under rotations. Math. Ann. 329, 685–706 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Brandolese L., Vigneron F.: New asymptotic profiles on nonstationary solutions of the Navier–Stokes system. J. Math. Pures Appl. 88, 64–86 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Carpio A.: Large time behavior in the incompressible Navier–Stokes equations. SIAM J. Math. Anal. 27, 449–475 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Choe H.J., Bum J.J.: Decay rate of solutions of Navier–Stokes equations in 3-dimensional half-space. J. Math. Anal. Appl. 339, 1485–1496 (2007)

    Article  Google Scholar 

  18. Choe H.J., Jin B.J.: Weighted estimates of the asymptotic profiles of the Navier–Stokes flow in \({\mathbb{R}^n}\). J. Math. Anal. Appl. 344, 353–366 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Deuring, P.: An integral operator related to the Stokes system in exterior domains. Math. Methods Appl. Sci. 13, 323–333 (1990) [Addendum. Math. Methods Appl. Sci. 14, 445 (1991)]

  20. Deuring P.: The resolvent problem for the Stokes system in exterior domains: an elementary approach. Math. Methods Appl. Sci. 13, 335–349 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Deuring P.: The Stokes system in exterior domains: L p-estimates for small values of a resolvent parameter. J. Appl. Math. Phys. (ZAMP) 41, 829–842 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  22. Deuring P.: Spatial decay of time-dependent Oseen flows. SIAM J. Math. Anal. 41, 886–922 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Deuring P.: A representation formula for the velocity part of 3D time-dependent Oseen flows. J. Math. Fluid Mech. 16, 1–39 (2014)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. 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)

    Article  MATH  MathSciNet  Google Scholar 

  25. Deuring P.: Spatial decay of time-dependent incompressible Navier–Stokes flows with nonzero velocity at infinity. SIAM J. Math. Anal. 45, 1388–1421 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Friedman A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  27. Friedman A.: Partial Differential Equations. Reinhart and Winston, Holt (1969)

    MATH  Google Scholar 

  28. Fujigaki Y., Miyakawa T.: Asymptotic profiles of the nonstationary incompressible Navier–Stokes flows in \({\mathbb{R}^{n}}\). SIAM J. Math. Anal. 33, 523–544 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  29. Galdi G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. I. Linearised Steady Problems (corr. 2nd print.). Springer, New York (1998)

    Google Scholar 

  30. Giga Y.: Analyticity of the semigroup generated by the Stokes operator in L r spaces. Math. Z. 178, 297–321 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  31. Giga Y., Sohr H.: Abstract L p-estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  32. Golovkin, K.K., Ladyženskaya, O.A.: Solutions of nonstationary boundary value problems for the Navier–Stokes equations. Trudy Mat. Inst. Steklov. 59, 100–114 (1960) (Russian)

  33. He C., Hsiao L.: The decay rates of strong solutions for the Navier–Stokes equations. J. Math. Anal. Appl. 268, 417–425 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  34. He C., Miyakawa T.: On L 1-summability and asymptotic profiles for smooth solutions to Navier–Stokes equations in 3D exterior domain. Math. Z. 238, 799–816 (2001)

    MathSciNet  Google Scholar 

  35. He C., Miyakawa T.: On weighted-norm estimates for incompressible Navier–Stokes flows in 3D exterior domains. J. Differ. Equ. 246, 2355–2386 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  36. He C., Xin Z.: On the decay properties of solutios to the nonstationary Navier–Stokes equations in \({\mathbb{R}^3}\). Proc. R. Soc. Edinb. Sect. A 131, 587–619 (2001)

    Article  MathSciNet  Google Scholar 

  37. Knightly G.H.: On a class of global solutions of the Navier–Stokes equations. Arch. Rat. Mech. Anal. 21, 211–245 (1996)

    MathSciNet  Google Scholar 

  38. Knightly G.H.: A Cauchy problem for the Navier–Stokes equations in \({\mathbb{R}^{n}}\). SIAM J. Math. Anal. 3, 506–511 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  39. Kozono H.: Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains. Math. Ann. 320, 709–730 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  40. Kozono H.: Global L n-solution and its decay property for the Navier–Stokes equations in half-space in \({\mathbb{R}_{+}^{n}}\). J. Differ. Equ. 79, 79–88 (1989)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  41. Kukavica I.: Space-time decay for solutions of the Navier–Stokes equations. Indiana Univ. Math. J. 50, 205–222 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  42. Kukavica I., Torres J.J.: Weighted L p-decay for solutions of the Navier–Stokes equations. Commun. Partial Differ. Equ. 32, 819–831 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  43. Lemarié-Rieusset P.G.: Recent Developments in the Navier–Stokes Problem. Chapman & Hall, CRC Press (2002)

    Book  MATH  Google Scholar 

  44. Maremonti P.: Partial regularity of a generalized solution to the Navier–Stokes equations in exterior domain. Commun. Math. Phys. 110, 75–87 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  45. Maremonti P.: On the asymptotic behaviour of the L 2-norm of suitable weak solutions to the Navier–Stokes equations in three-dimensional exterior domains. Commun. Math. Phys. 118, 385–400 (1988)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  46. Maremonti P.: Some results on the asymptotic behavior of Hopf weak solutions to the Navier–Stokes equations in unbounded domains. Math. Z. 210, 1–22 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  47. Maremonti P., Solonnikov V.A.: On nonstationary Stokes problem in exterior domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24, 395–449 (1997)

    MATH  MathSciNet  Google Scholar 

  48. Miyakawa T.: On space time decay properties of nonstationary incompressible Navier–Stokes flows in \({\mathbb{R} ^n}\). Funkcial. Ekvac. 43, 541–557 (2000)

    MATH  MathSciNet  Google Scholar 

  49. Miyakawa T., Schonbek M.E.: On optimal decay rates for weak solutions to the Navier–Stokes equations. Math. Bohemica 126, 443–455 (2001)

    MATH  MathSciNet  Google Scholar 

  50. Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  MATH  Google Scholar 

  51. Shen Z.: 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)

    Article  MATH  Google Scholar 

  52. Schonbek M.E.: L 2-decay for weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 88, 209–222 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  53. Schonbek M.E.: Lower bounds of rates of decay for solutions to the Navier–Stokes equations. J. Am. Math. Soc. 4, 423–499 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  54. Schonbek M.E., Schonbek T.P.: On the boundedness and decay of moments of solutions to the Navier–Stokes equations. Adv. Differ. Equ. 5, 861–898 (2000)

    MATH  MathSciNet  Google Scholar 

  55. Skalak Z.: Fast decays of strong global solutions of the Navier–Stokes equations. J. Math. Fluid Mech. 9, 565–587 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  56. Sohr H.: The Navier–Stokes Equations. An Elementary Functional Analytic Approach. Birkhäuser, Basel (2001)

    MATH  Google Scholar 

  57. Solonnikov, V.A.: A priori estimates for second order parabolic equations. Trudy Mat. Inst. Steklov. 70, 133–212 (1964) (Russian) [English translation, AMS Translations 65, 51–137 (1967)]

  58. Solonnikov, V.A.: Estimates for solutions of nonstationary Navier–Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38, 153–231 (1973) (Russian) [English translation, J. Soviet Math. 8, 467–529 (1977)]

  59. Takahashi S.: A weighted equation approach to decay rate estimates for the Navier–Stokes equations. Nonlinear Anal. 37, 751–789 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  60. Teman R.: Navier–Stokes Equations. Theory and Numerical Analysis. AMS Chelsea Publishing, Providence (2001)

    Google Scholar 

  61. Vigneron F.: Spatial decay of the velocity field of an incompressible viscous fluid in \({\mathbb{R}^{d}}\). Nonlinear Anal. 63, 525–549 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  62. Wiegner M.: Decay results for weak solutions of the Navier–Stokes equations in \({\mathbb{R}^{n}}\). J. Lond. Math. Soc. 35, 307–313 (1987)

    MathSciNet  Google Scholar 

  63. von Wahl, W.: Vorlesung über das Außenraumproblem für die instationären Gleichungen von Navier–Stokes. Rudolph-Lipschitz-Vorlesung 11, Rheinische Friedrich-Wilhelms-Universität Bonn (1989)

  64. Yoshida K.: Functional Analysis, 6th edn. Springer, Berlin (1980)

    Book  Google Scholar 

  65. Zheng S.: Asymptotic behavior for strong solutions of the Navier–Stokes equations with external forces. Nonlinear Anal. 45, 435–446 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Univ Lille Nord de France, 59000, Lille, France

    Paul Deuring

  2. ULCO, LMPA, CS 80699, 62228, Calais Cedex, France

    Paul Deuring

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Communicated by G. P. Galdi

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Deuring, P. Pointwise Spatial Decay of Weak Solutions to the Navier–Stokes System in 3D Exterior Domains. J. Math. Fluid Mech. 17, 199–232 (2015). https://doi.org/10.1007/s00021-014-0198-x

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