Advertisement

On Schauder Estimates for the Evolution Generalized Stokes Problem

  • Vsevolod A. Solonnikov
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

This note is devoted to coercive estimates in anisotropic Hölder norms of the solution of the Cauchy—Dirichlet problem for the system of generalized Stokes equations arising in the linearization of equations of motion of a certain class of non-Newtonian liquids.

Keywords

Dirichlet Problem English Transl Stokes Problem Exterior Domain Stokes System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Amann, Stability of the rest state of a viscous incompressible fluid, Arch. Rat. Mech. Anal. 126 (1994) n. 3, 231–242.CrossRefMathSciNetGoogle Scholar
  2. [2]
    V.S. Belonosov, Estimates of the solutions of parabolic systems in Holder weight classes and some of their applications (Russian), Mat. Sb. 110 (1979) n. 2, 163–188; English transl. in Math. USSR-Sb. 38 (1981).MathSciNetGoogle Scholar
  3. [3]
    V.S. Belonosov and T.I. Zeleniak, Nonlocal problems in the theory of quasilinear parabolic equations (Russian), “Nauka”, Novosibirsk, 1975.Google Scholar
  4. [4]
    G.I. Bizhanova, Solution in a weighted Hölder space of an initial-boundary value problem for a second order parabolic equation with a time derivative in the conjugation condition (Russian), Algebra i Analiz 6 (1994) n. 1, 64–94; English transl. in St. Petersburg Math. J. 6 (1995) n. 1, 51-75.MathSciNetGoogle Scholar
  5. [5]
    G.I. Bizhanova and V.A. Solonnikov, On the solvability of an initial-boundary value problem for a second order parabolic equation with a time derivative in the boundary condition in a weighted Hölder space of functions (Russian), Algebra i Analiz 5 (1993) n. 1, 109–142; English transl. in St. Petersburg Math. J., 5 (1994) n. 1, 97-124.MathSciNetGoogle Scholar
  6. [6]
    M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math. 330 (1982), 173–214.MathSciNetGoogle Scholar
  7. [7]
    A.G. Hachiatrian and V.A. Solonnikov, Estimates for solutions of parabolic initialboundary value problems in weighted Holder norms (Russian), Trudy Mat. Inst. Steklov 147 (1980), 147–155; English transl. in Proc. Steklov Inst. Mat. (1981), n. 2.MathSciNetGoogle Scholar
  8. [8]
    O.A. Ladyzhenskaya and G.A. Seregin, Coercive estimates for solutions of linearizations of modified Navier-Stokes equations (Russian), Dokl. Acad. Nauk 370 (2000) n. 6, 738–740; English transl., Doklady Mathematics 61 (2000) n. 1, 113-115.MathSciNetGoogle Scholar
  9. [9]
    O.A. Ladyzhenskaya, On multiplicators in Holder spaces with nonhomogeneous metric, Methods Appl. Anal. 7 (2000) n. 3, 465–472.MathSciNetGoogle Scholar
  10. [10]
    J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and measure-valued solutions to evolutionary partial differential equations, Applied mathematics and mathematical computations, 13, Chapman & Hall, London, 1996.Google Scholar
  11. [11]
    J. Málek, J. Nečas and M. Růžička, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p = 2, Adv. Differential Equations 6 (2001) n. 3, 257–302.MathSciNetGoogle Scholar
  12. [12]
    V.A. Solonnikov, On the differential properties of the solution of the first boundary value problem for a non-stationary system of Navier-Stokes equations (Russian), Trudy Mat. Inst. Steklov: 73 (1964), 222–291.MathSciNetGoogle Scholar
  13. [13]
    V.A. Solonnikov, Estimates of solutions of non-stationary Navier-Stokes equations (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 153–231; English transl., J. Soviet Math. 8 (1977) n. 4, 467-529.MathSciNetGoogle Scholar
  14. [14]
    V.A Solonnikov, On estimates of maxima moduli of the derivatives of the solution of uniformly parabolic initial-boundary value problem (Russian), LOMI Preprint P-2-77 (1977), 3–20.Google Scholar
  15. [15]
    V.A Solonnikov, L p-estimates for solutions to the initial boundary value problem for the generalized Stokes system in a bounded domain. Function theory and partial differential equations (Russian), Problemy Mat. Anal. 21 (2000), 211–263; English transl., J. Math. Sci. 105 (2001) n. 5, 2448-2484.Google Scholar
  16. [16]
    V.A. Solonnikov, An initial boundary value problem for a generalized system of Stokes equations in a half-space (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), 224–275; English transl. in J. Math. Sci. 115 (2003) n. 6, 2832-2861.Google Scholar
  17. [17]
    V.A. Solonnikov, Model problem for n-dimensional generalized Stokes equations, Nonl. Anal. 47 (2001) n. 6, 4139–4150.CrossRefMathSciNetGoogle Scholar
  18. [18]
    V.A. Solonnikov, Schauder estimates for evolution generalized Stokes problem, PDMI Preprint 25 (2005), 1–40 (electronic version at: http://www.pdmi.ras.ru/).Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Vsevolod A. Solonnikov
    • 1
  1. 1.Department of MathematicsUniversity of FerraraFerraraItaly

Personalised recommendations