Journal of Mathematical Sciences

, Volume 195, Issue 1, pp 20–60 | Cite as

The linearization principle for a free boundary problem for viscous, capillary incompressible fluids

  • S. J. N. Mosconi
  • V. A. Solonnikov

The free boundary problem associated with a viscous incompressible surface wave subject to the capillary force on a free upper surface and the Dirichlet boundary condition on a fixed bottom surface is considered. In the spatially periodic case, a general linearization principle is proved, which gives, for sufficiently small perturbations from a linearly stable stationary solution, the existence of a global solution of the associated system and the exponential convergence of the latter to the stationary one. The convergence of the velocity, the pressure, and the free boundary is proved in anisotropic Sobolev--Slobodetskii spaces, and then a suitable change of variables is performed to state the problem in a fixed domain. This linearization principle is applied to the study of the rest state's stability in the case of general potential forces. Bibliography: 16 titles.


Surface Wave Free Boundary Dirichlet Boundary Bottom Surface Global Solution 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. S. Agranovich and M. I. Vishik, ``Elliptic problems with a parameter and parabolic problems of general type,'' Uspekhi Mat. Nauk, 19, No. 3, 53--161 (1964).MATHGoogle Scholar
  2. 2.
    J. T. Beale, ``The initial value problem for the Navier--Stokes equations with a free boundary,'' Comm. Pure Appl. Math., 31, 359--392 (1980).Google Scholar
  3. 3.
    J. T. Beale, ``Large-time regularity of viscous surface waves,'' Arch. Rat. Mech. Anal., 84, 307--352 (1984).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    J. T. Beale and T. Nishida, ``Large-time behaviour of viscous surface waves,'' Lect. Notes Num. Appl. Anal., 8, 1--14 (1985).MathSciNetGoogle Scholar
  5. 5.
    M. E. Bogovskii, ``Resolution of some problems of the vector analysis connected with the operators div and grad,'' Trudy S. L. Sobolev Semin., 1, 5--40 (1980).MathSciNetGoogle Scholar
  6. 6.
    O. A. Ladyzenskaya and V. A. Solonnikov, ``The linearization principle and invariant manifolds for problems in magneto-hydrodynamics,'' J. Math. Sci., 8, 384--422 (1977).CrossRefGoogle Scholar
  7. 7.
    T. Nishida, Y. Teramoto, and H. Yoshihara, ``Global in time behaviour of viscous surface waves: horizontally periodic motion,'' J. Math. Kyoto Univ., 44, 271--323 (2004).MathSciNetMATHGoogle Scholar
  8. 8.
    M. Padula and V. A. Solonnikov, ``On Rayleigh--Taylor stability,'' Ann. Univ. Ferrara, Sez. VIII sc. Mat., 46, 307--336 (2000).MathSciNetMATHGoogle Scholar
  9. 9.
    V. A. Solonnikov, ``On the stability of uniformly rotating viscous incompressible self-gravitating liquid,'' Zap. Nauchn. Semin. POMI, 348, 165--208 (2007).MathSciNetGoogle Scholar
  10. 10.
    V. A. Solonnikov, ``On the stability of uniformly rotating viscous incompressible self-gravitating liquid,'' J. Math. Sci., 152, No. 5, 4343--4370 (2008).Google Scholar
  11. 11.
    V. A. Solonnikov, ``On the linear problem arising in the study of a free boundary problem for the Navier--Stokes equation,'' Algebra Analiz, 22, No. 6, 235--269 (2010).MathSciNetGoogle Scholar
  12. 12.
    D. Sylvester, ``Large time existence of small viscous surface waves without surface tension,'' Comm. Partial Diff. Eqs., 15, 823--903 (1990).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    13.N. Tanaka and A. Tani, ``Large-time existence of surface waves in incompressible viscous fluid with or without surface tension,'' Arch. Rat. Mech. Anal., 130, 303--314 (1995).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    A. Tani, ``Small-time existence for the three-dimensional incompressible Navier--Stokes equations with a free surface,'' Arch. Rat. Mech. Anal., 133, 299--331 (1996).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Y. Teramoto, ``On the Navier--Stokes flow down an inclined plane,'' J. Math. Kyoto Univ., 32, 593--619 (1992).MathSciNetMATHGoogle Scholar
  16. 16.
    Y. Teramoto, ``The initial value problem for a viscous incompressible flow down an inclined plane,'' Hiroshima Math. J., 15, 619--643 (1985).MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.University of CataniaCataniaItaly
  2. 2.St.Petersburg Department of the Steklov Mathematical InstituteSt.PetersburgRussia

Personalised recommendations