Skip to main content
SpringerLink
Log in

Resolvent Estimates in L p for the Stokes Operator in Lipschitz Domains

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We establish the L p resolvent estimates for the Stokes operator in Lipschitz domains in \({\mathbb{R}^d}\), \({d\geqq 3}\) for \({|\frac{1}{p}-\frac{1}{2}| < \frac{1}{2d} +\varepsilon}\). The result implies that the Stokes operator in a three-dimensional Lipschitz domain generates a bounded analytic semigroup in L p for (3/2) − ε < p < 3 + ε. This gives an affirmative answer to a conjecture of M. Taylor (Progr. Nonlinear Differential Equations Appl., vol. 42, pp. 320–334).

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. Brown R.: The method of layer potentials for the heat equation in Lipschitz cylinders. Am. J. Math. 111, 339–379 (1989)

    Article  MATH  Google Scholar 

  2. Brown R., Shen Z.: Estimates for the Stokes operator in Lipschitz domains. Indiana Univ. Math. J. 44, 1183–1206 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Caffarelli L., Peral I.: On W 1,p estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51, 1–21 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Coifman R., McIntosh A., Meyer Y.: L’intégrale de Cauchy définite un opèrateur bornè sur L 2 pour les courbes lipschitziennes. Ann. Math. 116, 361–387 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dahlberg B.: Estimates of harmonic measure. Arch. Rational Mech. Anal. 65, 275–288 (1977)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Dahlberg B., Kenig C., Verchota G.: Boundary value problems for the system of elastostatics in Lipschitz domains. Duke Math. J. 57(3), 795–818 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Deuring P.: The Stokes resolvent in 3D domains with conical boundary points: non-regularity in L p-spaces. Adv. Differ. Equ. 6, 175–228 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Deuring P., von Wahl W.: Strong solutions of the Navier–Stokes system on Lipschitz bounded domains. Math. Nachr. 171, 111–148 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fabes E., Kenig C., Verchota G.: The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J. 57(3), 769–793 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fabes E., Mendez O., Mitrea M.: Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159, 323–368 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fujita H., Kato T.: On the Navier–Stokes initial value problem I. Arch. Rational Mech. Anal. 16, 269–315 (1964)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Fujiwara D., Morimoto H.: An L r -theorem for the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo Sect. I-A Math. 24, 685–700 (1977)

    MathSciNet  MATH  Google Scholar 

  14. Geng, J., Kilty, J.: The L p regularity problem for the Stokes system on Lipschitz domains. Preprint (2011)

  15. Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Ann. Math. Studies, vol. 105. Princeton University Press, 1983

  16. Giga Y.: Analyticity of the semigroup generated by the Stokes operator in L r spaces. Math. Z. 178, 297–329 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  17. Giga Y., Miyakawa T.: Solutions in L r of the Navier–Stokes initial value problem. Arch. Rational Mech. Anal. 89, 267–281 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Jerison D., Kenig C.: An identity with applications to harmonic measure. Bull. Am. Math. Soc. 2, 447–451 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jerison D., Kenig C.: The Neumann problem in Lipschitz domains. Bull. Am. Math. Soc. (N.S.) 4, 203–207 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kenig, C., Lin, F., Shen, Z.: Homogenization of elliptic systems with Neumann boundary conditions. Preprint. arXiv:1010.6114 (2010)

  21. Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, 1963

  22. Lebedev, N.N.: Special Functions and Their Applications. Translated and edited by R. Silverman. Dover, 1972

  23. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, 2000

  24. Mitrea M., Monniaux S.: The regularity of the Stokes operator and the Fujita-Kato approach to the Navier–Stokes initial value problem in Lipschitz domains. J. Funct. Anal. 254, 1522–1574 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mitrea M., Monniaux S.: On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian Manifolds. Trans. Am. Math. Soc. 361, 3125–3157 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mitrea, M., Wright, M.: Boundary value problems for the Stokes system in arbitrary Lipschitz domains. Astérisque (to appear)

  27. Shen Z.: Boundary value problems for parabolic Lame systems and a non stationary linearized system of Naver-Stokes equations in Lipschitz cylinders. Am. J. Math. 113, 293–373 (1991)

    Article  MATH  Google Scholar 

  28. Shen Z.: Resolvent estimates in L p for elliptic systems in Lipschitz domains. J. Funct. Anal. 133, 224–251 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shen Z.: Bounds of Riesz transforms on L p spaces for second order elliptic operators. Ann. Inst. Fourier (Grenoble) 55, 173–197 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shen Z.: Necessary and sufficient conditions for the solvability of the L p Dirichlet problem on Lipschitz domains. Math. Ann. 336(3), 697–724 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Solonnikov V.A.: Estimates for solutions of nonstationary Navier–Stokes equations. J. Soviet Math. 8, 467–529 (1977)

    Article  MATH  Google Scholar 

  32. Stein E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  33. Taylor, M.: Incompressible fluid flow on rough domains. Semigroups of Operators: Theory and Applications (Newport Beach, CA, 1998). Progr. Nonlinear Differential Equations Appl., vol. 42, pp. 320–334. Birkhäuser, Basel, 2000

  34. Verchota G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  35. von Wahl, W.: The Equations of Navier–Stokes and Abstract Parabolic Equations. Vieweg, Braunschweig, 1985

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, Lexington, KY, 40506, USA

    Zhongwei Shen

Authors
  1. Zhongwei Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhongwei Shen.

Additional information

Communicated by F. Lin

Supported in part by NSF grant DMS-0855294.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shen, Z. Resolvent Estimates in L p for the Stokes Operator in Lipschitz Domains. Arch Rational Mech Anal 205, 395–424 (2012). https://doi.org/10.1007/s00205-012-0506-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-012-0506-7

Keywords

Search

Navigation