On the Equation div u = g and Bogovskii’s Operator in Sobolev Spaces of Negative Order

  • Matthias Geißert
  • Horst Heck
  • Matthias Hieber
Part of the Operator Theory: Advances and Applications book series (OT, volume 168)


Consider the divergence problem with homogeneous Dirichlet data on a Lipschitz domain. Two approaches for its solutions in the scale of Sobolev spaces are presented. The first one is based on Calderón-Zygmund theory, whereas the second one relies on the Stokes equation with inhomogeneous data.


Divergence problem Bogovskii’s operator 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M.E. Bogovskii, Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Dokl. Akad. Nauk SSSR. 248 (1979), (5), 1037–1040.Google Scholar
  2. [2]
    M.E. Bogovskii, Solution of some vector analysis problems connected with operators div and grad. Trudy Seminar S. Sobolev, No. 1, 1980, Akademia Nauk SSSR, Sibirskoe Otdelnie Matematiki, Novosibirsk, (1980), 5–40.Google Scholar
  3. [3]
    W. Borchers and H. Sohr, On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J. 19 (1990), (1), 67–87.zbMATHMathSciNetGoogle Scholar
  4. [4]
    L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova 31 (1961), 308–340.zbMATHMathSciNetGoogle Scholar
  5. [5]
    P. Clément, and S. Li, Abstract parabolic quasilinear equations and application to a groundwater flow problem, Adv. Math. Sci. Appl., 3 (1993/94), 17–32.MathSciNetGoogle Scholar
  6. [6]
    A.P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains, J. Math. Soc. Japan 46 (1994), (4), 607–643.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer, 1994, Linearized steady problems.Google Scholar
  9. [9]
    M. Geissert, H. Heck, and M. Hieber, L p-theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle, to appear in J. Reine Angew. Math.Google Scholar
  10. [10]
    G.P. Galdi and C.G. Simader, Existence, uniqueness and L q-estimates for the Stokes problem in an exterior domain. Arch. Ration. Mech. Anal. 112 (1990), (4), 291–318.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    H. Kozono and H. Sohr, New a priori estimates for the Stokes equations in exterior domains. Indiana Univ. Math. J. 40 (1991), (1), 1–27.zbMATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, 1969.Google Scholar
  13. [13]
    D. Mitrea and M. Mitrea, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in Sobolev-Besov spaces on nonsmooth domains. Preprint, 2005.Google Scholar
  14. [14]
    J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Paris, 1967.Google Scholar
  15. [15]
    K. Pileckas, Three-dimensional solenoidal vectors. J. Soviet Math. 21 (1983), 821–823.zbMATHCrossRefGoogle Scholar
  16. [16]
    H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach. Birkhäuser, 2001.Google Scholar
  17. [17]
    V.A. Solonnikov, Stokes and Navier-Stokes equations in domains with noncompact boundaries, In: Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. IV (Paris, 1981/1982), Res. Notes in Math., Vol. 84, Pitman, 1983, 240–349.MathSciNetGoogle Scholar
  18. [18]
    V.A. Solonnikov and V.E. Ščadilov, On a boundary value problem for stationary Navier-Stokes equations, Proc. Steklov Math. Inst. 125 (1973), 186–199.zbMATHGoogle Scholar
  19. [19]
    E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.zbMATHGoogle Scholar
  20. [20]
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd ed. Johann Ambrosius Barth, Heidelberg, Leipzig, 1995.zbMATHGoogle Scholar
  21. [21]
    W. von Wahl, On necessary and sufficient conditions for the solvability of the equations rot u = γ and div u = ε with u vanishing on the boundary, The Navier-Stokes equations (Oberwolfach, 1988), Springer Lecture Notes in Math 1431, 1990, 152–157.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Matthias Geißert
    • 1
  • Horst Heck
    • 1
  • Matthias Hieber
    • 1
  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

Personalised recommendations