Bounded Imaginary Powers and H∞-Calculus of the Stokes Operator in Unbounded Domains

  • H. Abels
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 64)

Abstract

In the present contribution we study the Stokes operator A q = -P q Δ on Lσq (Ω), 1 < q < ∞, where Ω is a suitable bounded or unbounded domain in ℝn, n ≥ 2, with C1,1-boundary. We present some conditions on Ω and the related function spaces and basic equations which guarantee that c + A q for suitable c ∈ ℝ is of positive type and admits a bounded H∞- calculus. This implies the existence of bounded imaginary powers of c + A q . Most domains studied in the theory of Navier-Stokes like, e.g., bounded, exterior, and aperture domains as well as asymptotically flat layers satisfy the conditions. The proof is done by constructing an approximate resolvent based on the results of [3], which were obtained by applying the calculus of pseudodifferential boundary value problems. Finally, the result is used to proof the existence of a bounded H∞-calculus of the Stokes operator Aq on an aperture domain.

Keywords

Stokes equations exterior domains bounded imaginary powers H∞-calculus aperture domain 
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References

  1. [1]
    H. Abels. Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients. Preprint No. 2343, TU Darmstadt, 2004, to appear in Comm. Part. Diff. Eq.Google Scholar
  2. [2]
    H. Abels. Reduced and generalized Stokes resolvent equations in asymptotically flat layers, part I: unique solvability. J. Math. Fluid. Mech. 7, No. 2, 201–222, 2005.MATHMathSciNetGoogle Scholar
  3. [3]
    H. Abels. Reduced and generalized Stokes resolvent equations in asymptotically flat layers, part II: H in∞-calculus. J. Math. Fluid. Mech. 7, No. 2, 223–260, 2005.MATHMathSciNetGoogle Scholar
  4. [4]
    H. Abels. Bounded imaginary powers of the Stokes operator in an infinite layer. J. Evol. Eq. 2, 439–457, 2002.MATHMathSciNetGoogle Scholar
  5. [5]
    H. Abels. Bounded Imaginary Powers and H∞-Calculus of the Stokes Operators in Two-Dimensional Exterior Domains. Preprint No. 2338, TU Darmstadt, 2004, to appear in Math. Z.Google Scholar
  6. [6]
    H. Abels and M. Wiegner. Resolvent estimates for the Stokes operator on an infinite layer. Preprint TU Darmstadt, 2004 to appear in Diff. Int. Eq.Google Scholar
  7. [7]
    R.A. Adams. Sobolev Spaces. Academic Press, New York, San Francisco, London, 1975.Google Scholar
  8. [8]
    H. Amann, M. Hieber, and G. Simonett. Bounded H∞-calculus for elliptic operators. Diff. Int. Eq., Vol. 7, No. 3, 613–653, 1994.MathSciNetGoogle Scholar
  9. [9]
    W. Borchers and H. Sohr. On the semigrup of the Stokes operator for exterior domains in Lq-spaces. Math. Z. 196, 415–425, 1987.CrossRefMathSciNetGoogle Scholar
  10. [10]
    S.K. Chua. Extension theorems on weighted Sobolev space. Indiana Univ. Math. J., 41, 1027–1076, 1992.CrossRefMATHMathSciNetGoogle Scholar
  11. [11]
    R. Denk, M. Hieber, and J. Prüss. R-boundedness, Fourier multipliers and problems of elliptic and parabolic Type. Mem. Am. Math. Soc. 788, 114 p., 2003.Google Scholar
  12. [12]
    W. Desch, M. Hieber, and J. Prüss. Lp-theory of the Stokes equation in a half-space. J. Evol. Eq. 1, 115–142, 2001.Google Scholar
  13. [13]
    G. Dore and A. Venni. On the closedness of the sum of two closed operators. Math. Z. 196, 189–201, 1987.CrossRefMathSciNetGoogle Scholar
  14. [14]
    R. Farwig. Weighted Lq-Helmholtz decompositions in infinite cylinders and in infinite layers. Adv. Diff. Eq. 8, 357–384, 2003.MATHMathSciNetGoogle Scholar
  15. [15]
    R. Farwig and H. Sohr. Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Japan 46, No. 4, 607–643, 1994.MathSciNetGoogle Scholar
  16. [16]
    R. Farwig and H. Sohr. Helmholtz Decomposition and Stokes Resolvent System for Aperture Domains in Lq-Spaces. Analysis 16, 1–26, 1996.MathSciNetGoogle Scholar
  17. [17]
    M. Franzke. Strong Lq-theory of the Navier-Stokes equations in aperture domains. Ann. Univ. Ferrara, Nuova Ser., Sez. VII 46, 161–173, 2000.MATHMathSciNetGoogle Scholar
  18. [18]
    A. Fröhlich. Maximal regularity for the non-stationary Stokes system in an aperture domain. J. Evol. Eq. 2, 471–493, 2002.MATHGoogle Scholar
  19. [19]
    Y. Giga. Analyticity of the semigroup generated by the Stokes operator in Lr spaces. Math. Z. 178, 297–329, 1981.CrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    Y. Giga. Domains of fractional powers of the Stokes operator in Lr Spaces. Arch. Ration. Mech. Anal. 89, 251–265, 1985.MATHMathSciNetGoogle Scholar
  21. [21]
    Y. Giga and H. Sohr. On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36, 103–130, 1989.MathSciNetGoogle Scholar
  22. [22]
    Y. Giga and H. Sohr. Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94, 1991.CrossRefMathSciNetGoogle Scholar
  23. [23]
    G. Grubb. Functional Calculus of Pseudodifferential Boundary Problems, 2nd Edition. Birkhäuser, Basel-Boston-Berlin, 1996.Google Scholar
  24. [24]
    G. Grubb and V.A. Solonnikov. Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods. Math. Scand. 69, 217–290, 1991.MathSciNetGoogle Scholar
  25. [25]
    A. McIntosh. Operators which have an H∞-calculus. In “Miniconference on Operator Theory and Partial Differential Equations”, B. Jefferies, A. McIntosh, W. Ricker, editors Proc. Center Math. Anal. A.N.U., 14, 210–231, 1986.Google Scholar
  26. [26]
    T. Miyakawa. The Helmholtz decomposition of vector fields in some unbounded domains. Math. J. Toyama Univ. 17, 115–149, 1994.MATHMathSciNetGoogle Scholar
  27. [27]
    A. Noll and J. Saal. H∞-calculus for the Stokes operator on Lq-spaces. Math. Z. 244, 651–688, 2003.MathSciNetGoogle Scholar
  28. [28]
    C.G. Simader and H. Sohr. A new approach to the Helmholtz decomposition and the Neumann problem in Lq-spaces for bounded and exterior domains. Series on Advances in Mathematics for Applied Sciences, Vol. 11, Singapore, World Scientific, 1–35, 1992.Google Scholar
  29. [29]
    H. Triebel. Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publishing Company, Amsterdam, New York, Oxford, 1978.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • H. Abels
    • 1
  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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