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

We consider the free boundary problem associated to a viscous incompressible surface wave subjected to capillary force on the free upper surface and Dirchlet boundary condition on the fixed bottom surface. In the spatially periodic case, we prove a general linearization principle which gives, for sufficiently small perturbations from a linearly stable stationary solution, existence of a global solution of the associated system and exponential convergence of the latter to the stationary one. Convergence of the velocity, the pressure and the free boundary is proved in anisotropic Sobolev–Slobodetskii spaces, after a suitable change of variables is performed to formulate the problem in a fixed domain. We apply this linearization principle to the study of the rest state's stability in the case of general potential forces.