Advertisement

Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 1155–1172 | Cite as

On Weighted Estimates for the Stokes Flows, with Application to the Navier–Stokes Equations

  • Pigong Han
Article

Abstract

Weighted estimates on the Stokes flows are given by means of the Stokes solution formula in the half space, which can be regarded as a complement and improvement on the previous known results. There are two main difficulties: in weighted cases, usual \(L^q-L^r\) estimates for the Stokes flows do not work any more, and the projection operator becomes unbounded possibly. Finally, as an application, employing these weighted estimates on the Stokes solution, we establish some weighted decay results for the Navier–Stokes flows.

Keywords

Stokes flow Solution formula Weighted estimate Strong solution 

Mathematics Subject Classification

35Q35 35B40 75D05 76D07 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author expresses sincere thanks to the anonymous referees for many helpful suggestions and kind comments. This work was supported by NSFC under Grant No. 11471322; and NSFC-NRF under Grant No. 11611540331; supported by Key Laboratory of RCSDS, CAS (No. 2008DP173182); supported by Youth Innovation Promotion Association of the Chinese Academy of Sciences.

Compliance with ethical standards

Conflict of interest

The author declares that the work described in the above article complied with the ethical standards, has also not been submitted elsewhere for publication, in whole or in part. Moreover, there is no financial or competing interests to disclose in relation to this work.

References

  1. 1.
    Bae, H.: Temporal decays in \(L^1\) and \(L^\infty \) for the Stokes flow. J. Differ. Equ. 222, 1–20 (2006)ADSCrossRefMATHGoogle Scholar
  2. 2.
    Bae, H.: Temporal and spatial decays for the Stokes flow. J. Math. Fluid Mech. 10, 503–530 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bae, H.: Analyticity and asymptotics for the Stokes solutions in a weighted space. J. Math. Anal. Appl. 269, 149–171 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bae, H., Choe, H.: Decay rate for the incompressible flows in half spaces. Math. Z. 238, 799–816 (2001)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bae, H., Jin, B.: Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains. J. Funct. Anal. 240, 508–529 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bae, H., Jin, B.: Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains. Bull. Korean Math. Soc. 44, 547–567 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bae, H., Jin, B.: Upper and lower bounds of temporal and spatial decays for the Navier–Stokes equations. J. Differ. Equ. 209, 365–391 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bae, H., Jin, B.: Temporal and spatial decays for the Navier–Stokes equations. Proc. R. Soc. Edinb. Sect. A 135, 461–477 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bae, H., Jin, B.: 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)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Brandolese, L.: Space-time decay of Navier–Stokes flows invariant under rotations. Math. Ann. 329, 685–706 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Brandolese, L., Vigneron, F.: New asymptotic profiles of nonstationary solutions of the Navier–Stokes system. J. Math. Pures Appl. 88, 64–86 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Borchers, W., Miyakawa, T.: \(L^2\) decay for the Navier–Stokes flow in half spaces. Math. Ann. 282, 139–155 (1988)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Comm. Pure. Appl. Math. 35, 771–831 (1982)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 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)CrossRefMATHGoogle Scholar
  16. 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)MathSciNetMATHGoogle Scholar
  17. 17.
    Chae, D.: Liouville type theorems for the Euler and the Navier–Stokes equations. Adv. Math. 228, 2855–2868 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 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)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 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)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Chang, T., Jin, B.: Notes on the space-time decay rate of the Stokes flows in the half space. J. Differ. Equ. 263, 240–263 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Frohlich, A.: Solutions of the Navier–Stokes initial value problem in weighted \(L^q\)-spaces. Math. Nachr. 269/270, 150–166 (2004)CrossRefMATHGoogle Scholar
  22. 22.
    Fujigaki, Y., Miyakawa, T.: Asymptotic profiles of non stationary incompressible Navier–Stokes flows in the half-space. Methods Appl. Anal. 8, 121–158 (2001)MathSciNetMATHGoogle Scholar
  23. 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)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Han, P.: Asymptotic behavior for the Stokes flow and Navier–Stokes equations in half spaces. J. Differ. Equ. 249, 1817–1852 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Han, P.: Long-time behavior for the nonstationary Navier–Stokes flows in \(L^1\). J. Funct. Anal. 266, 1511–1546 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Han, P.: Large time behavior for the nonstationary Navier–Stokes flows in the half-space. Adv. Math. 288, 1–58 (2016)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    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)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    He, C., Wang, L.: Weighted \(L^p-\)estimates for Stokes flow in \(\mathbb{R}^n_+\) with applications to the non-stationary Navier–Stokes flow. Sci. China Math. 54, 573–586 (2011)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kobayashi, T., Kubo, T.: Weighted \(L^p-L^q\) estimates of Stokes semigroup in half-space and its application to the Navier–Stokes equations. In: Recent Developments of Mathematical Fluid Mechanics, pp. 337-349, Adv. Math. Fluid Mech., Birkh\(\ddot{a}\)user/Springer, Basel (2016)Google Scholar
  31. 31.
    Kobayashi, T., Kubo, T.: Weighted \(L^p\)-theory for the Stokes resolvent in some unbounded domains. Tsukuba J. Math. 37, 179–205 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Lin, F.: A new proof of the Caffarelli–Kohn–Nirenberg theorem. Commun. Pure Appl. Math. 51, 241–257 (1998)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Jin, B.: Weighted \(L^q-L^1\) estimate of the Stokes flow in the half space. Nonlinear Anal. 72, 1031–1043 (2010)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    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)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Schonbek, M.E.: \(L^2\) decay for weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 88, 209–222 (1985)CrossRefMATHGoogle Scholar
  36. 36.
    Schonbek, M.E.: Lower bounds of rates of decay for solutions to the Navier–Stokes equations. J. Am. Math. Soc. 4, 423–449 (1991)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Schonbek, M.E.: Asymptotic behavior of solutions to the three-dimensional Navier–Stokes equations. Indiana Univ. Math. J. 41, 809–823 (1992)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    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)CrossRefMATHGoogle Scholar
  39. 39.
    Schonbek, M.E.: The Fourier Splitting Method. Advances in Geometric Analysis and Continuum Mechanics. Int. Press, Cambridge, Stanford, CA (1995)Google Scholar
  40. 40.
    Schonbek, M.E.: Total variation decay of solutions to the Navier–Stokes equations. Methods Appl. Anal. 7, 555–564 (2000)MathSciNetMATHGoogle Scholar
  41. 41.
    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)MathSciNetCrossRefGoogle Scholar
  42. 42.
    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)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Stein, E.S.: Note on singular integrals. Proc. Am. Math. Soc. 8, 250–254 (1957)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Stein, E.S., Weiss, G.: Fractional integrals on \(n\)-dimensional Euclidean space. J. Math. Mech. 7, 503–514 (1958)MathSciNetMATHGoogle Scholar
  45. 45.
    Ukai, S.: A solution formula for the Stokes equation in \({\mathbb{R}}^N\). Commun. Pure Appl. Math. XL, 611–621 (1987)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

Personalised recommendations