Skip to main content
SpringerLink
Log in

Weighted Decay Results for the Nonstationary Stokes Flow and Navier–Stokes Equations in Half Spaces

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

The weighted L q−L q (q = 1,∞) estimates for the Stokes flow are given in half spaces. Further large-time weighted decays for the second spatial derivatives of the Navier–Stokes equations are established, where the unboundedness of the projection operator \({P: L^q(\mathbb{R}^n_+) \rightarrow L^q_\sigma(\mathbb{R}^n_+)}\) (q = 1,∞) is overcome by employing a decomposition for the convection term. The main results in this article are motivated by the work in Bae (J Differ Equ 222:1–20, 2006; J Math Fluid Mech 10:503–530, 2008) and Bae and Jin (Proc R Soc Edinb Sect A 135:461–477, 2005).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bae H.: Temporal decays in L 1 and L ∞ for the Stokes flow. J. Differ. Equ. 222, 1–20 (2006)

    Article  MATH  ADS  Google Scholar 

  2. Bae H.: Temporal and spatial decays for the Stokes flow. J. Math. Fluid Mech. 10, 503–530 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Bae H., Choe H.: Decay rate for the incompressible flows in half spaces. Math. Z. 238, 799–816 (2001)

    MATH  MathSciNet  Google Scholar 

  4. Bae H., Choe H.: A regularity criterion for the Navier–Stokes equations. Commun. Partial Differ. Equ. 32, 1173–1187 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bae H., Jin J.: Regularity for the Navier–Stokes equations with slip boundary condition. Proc. Am. Math. Soc. 136, 2439–2443 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bae H., Jin J.: Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains. J. Funct. Anal. 240, 508–529 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bae H., Jin J.: Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains. Bull. Korean Math. Soc. 44, 547–567 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Bae H., Jin J.: Existence of strong mild solution of the Navier–Stokes equations in the half space with nondecaying initial data. J. Korean Math. Soc. 49, 113–138 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Brandolese L.: On the localization of symmetric and asymmetric solutions of the Navier-Stokes equations in R n. C.R. Acad. Sci. Paris S’er. I Math. 332, 125–130 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure. Appl. Math. 35, 771–831 (1982)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Chae D.: Conditions on the pressure for vanishing velocity in the incompressible fluid flows in R N. Commun. Partial Differ. Equ. 37, 1445–1455 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Chae D.: On the Liouville type theorems with weights for the Navier–Stokes equations and Euler equations. Differ. Integral Equ. 25, 403–416 (2012)

    MATH  MathSciNet  Google Scholar 

  17. Chae D.: Liouville type theorems for the Euler and the Navier–Stokes equations. Adv. Math. 228, 2855–2868 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Chae D.: On the regularity conditions of suitable weak solutions of the 3D Navier–Stokes equations. J. Math. Fluid Mech. 12, 171–180 (2010)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Chae D.: On the a priori estimates for the Euler, the Navier–Stokes and the quasi-geostrophic equations. Adv. Math. 221, 1678–1702 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Fujigaki Y., Miyakawa T.: Asymptotic profiles of non stationary incompressible Navier–Stokes flows in the half-space. Methods Appl. Anal. 8, 121–158 (2001)

    MATH  MathSciNet  Google Scholar 

  21. Giga Y., Matsui S., Shimizu Y.: On estimates in Hardy spaces for the Stokes flow in a half space. Math. Z. 231, 383–396 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Galdi G.: An introduction to the mathematical theory of the Navier–Stokes equations, vol. I. Linearized Steady Problems, 38 edn . Springer, New York (1994)

    Google Scholar 

  23. Han P.: Weighted decay properties for the incompressible Stokes flow and Navier–Stokes equations in a half space. J. Differ. Equ. 253, 1744–1778 (2012)

    Article  MATH  ADS  Google Scholar 

  24. Han P.: Asymptotic behavior for the Stokes flow and Navier–Stokes equations in half spaces. J. Differ. Equ. 249, 1817–1852 (2010)

    Article  MATH  ADS  Google Scholar 

  25. Han P.: Decay results of solutions to the incompressible Navier–Stokes flows in a half space. J. Differ. Equ. 250, 3937–3959 (2011)

    Article  MATH  ADS  Google Scholar 

  26. Han P.: Weighted spatial decay rates for the Navier–Stokes flows in a half space. Proc. R. Soc. Edinb. Sect. A. 144, 491–510 (2014)

    Article  MATH  Google Scholar 

  27. He C., Wang L.: Moment estimates for weak solutions to the Navier–Stokes equations in half-space. Math. Methods Appl. Sci. 32, 1878–1892 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Lin F.: A new proof of the Caffarelli–Kohn–Nirenberg theorem. Commun. Pure Appl. Math. 51, 241–257 (1998)

    Article  MATH  Google Scholar 

  29. Jin B.: Weighted L q−L 1 estimate of the Stokes flow in the half space. Nonlinear Anal. 72, 1031–1043 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Jin B.: Spatial and temporal decay estimate of the Stokes flow of weighted L 1 initial data in the half space. Nonlinear Anal. 73, 1394–1407 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  32. Schonbek M.E.: Lower bounds of rates of decay for solutions to the Navier–Stokes equations. J. Am. Math. Soc. 4, 423–449 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  33. Schonbek M.E.: Asymptotic behavior of solutions to the three-dimensional Navier–Stokes equations. Indiana Univ. Math. J. 41, 809–823 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  34. Schonbek M.E.: Large time behaviour of solutions to the Navier–Stokes equations in H m spaces. Commun. Partial Differ. Equ. 20, 103–117 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  35. Schonbek M.E.: The Fourier splitting method. Advances in Geometric Analysis and Continuum Mechanics. International Press, Cambridge (1995)

    Google Scholar 

  36. Schonbek M.E.: Total variation decay of solutions to the Navier–Stokes equations. Methods Appl. Anal. 7, 555–564 (2000)

    MATH  MathSciNet  Google Scholar 

  37. Shimizu Y.: L ∞-estimate of first-order space derivatives of Stokes flow in a half space. Funk. Ekvac. 42, 291–309 (1999)

    MATH  Google Scholar 

  38. Solonnikov V.A.: Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces and estimates for the resolvent of the Stokes operator. Usp. Mat. Nauk. 58, 123–156 (2003)

    Article  MathSciNet  Google Scholar 

  39. Solonnikov V.A.: On nonstationary Stokes problem and Navier–Stokes problem in a half-space with initial data nondecreasing at infinity. J. Math. Sci. 114, 1726–1740 (2003)

    Article  MathSciNet  Google Scholar 

  40. Stein E.M.: Harmonic Analysis. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  41. Ukai S.: A solution formula for the Stokes equation in \({\mathbb{R}^N}\) . Commun. Pure Appl. Math. XL, 611–621 (1987)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Pigong Han

Authors
  1. Pigong Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pigong Han.

Additional information

Communicated by D. Chae

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, P. Weighted Decay Results for the Nonstationary Stokes Flow and Navier–Stokes Equations in Half Spaces. J. Math. Fluid Mech. 17, 599–626 (2015). https://doi.org/10.1007/s00021-015-0209-6

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00021-015-0209-6

Mathematics Subject Classification

Keywords

Search

Navigation