Generic properties of the spectrum of the Stokes system with Dirichlet boundary condition in $R^3$

Abstract

Let $(S{D}_{\Omega })$ be the Stokes operator defined in a bounded domain Ω of ${\mathbb{R}}^3$ with Dirichlet boundary conditions. We prove that, generically with respect to the domain Ω with $C^5$ boundary, the spectrum of $(S{D}_{\Omega })$ satisfies a non-resonant property introduced by C. Foias and J.C. Saut in [17] to linearize the Navier–Stokes system in a bounded domain Ω of ${\mathbb{R}}^3$ with Dirichlet boundary conditions. For that purpose, we first prove that, generically with respect to the domain Ω with $C^5$ boundary, all the eigenvalues of $(S{D}_{\Omega })$ are simple. That answers positively a question raised by J.H. Ortega and E. Zuazua in [27, Section 6]. The proofs of these results follow a standard strategy based on a contradiction argument requiring shape differentiation. One needs to shape differentiate at least twice the initial problem in the direction of carefully chosen domain variations. The main step of the contradiction argument amounts to study the evaluation of Dirichlet-to-Neumann operators associated to these domain variations.