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

Let (SDΩ) be the Stokes operator defined in a bounded domain Ω of R with Dirichlet boundary conditions. We first prove that, generically with respect to the domain Ω, all the eigenvalues of (SDΩ) are simple. That answers positively a question raised by J. H. Ortega and E. Zuazua in [18, Section 6]. Our second result states that, generically with respect to the domain Ω, the spectrum of (SDΩ) verifies a non resonant property introduced by C. Foias and J. C. Saut in [10] and used to linearize the Navier-Stokes system in a bounded domain Ω of R with Dirichlet boundary conditions. The proofs of these results follow a standard strategy based on a contradiction argument requiring shape differentiation. However, one needs to shape differentiate 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.