On linearization principle for Navier-Stokes system with dynamic slip boundary condition

Abstract

We consider incompressible Navier-Stokes equations in a bounded 3D regular domain, coupled with the so-called dynamic boundary condition. We rigorously establish the principle of linearized stability. Namely, given a smooth stationary state, we prove that the linearized equation has a complete basis of generalized eigenfunctions, and that the non-linear stability depends on the supremum of the real part of spectrum in the usual way. We work in the class of weak solutions satisfying the energy inequality, for which the global existence (but not uniqueness) is known.