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


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 
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© 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

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