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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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
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