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

TL;DR

This paper demonstrates that for a generic class of three-dimensional domains with smooth boundaries, the spectrum of the Stokes operator exhibits simple eigenvalues and a non-resonant property, aiding in the linearization of Navier-Stokes equations.

Contribution

It proves that generically, the eigenvalues of the Stokes operator are simple and satisfy a non-resonant property for domains with C^5 boundary, answering a previously open question.

Findings

Eigenvalues of the Stokes operator are generically simple.

The spectrum satisfies a non-resonant property.

Results are obtained via shape differentiation and domain variation analysis.

Abstract

Let (SD_\Omega) be the Stokes operator defined in a bounded domain \Omega of R^3 with Dirichlet boundary conditions. We prove that, generically with respect to the domain \Omega with C^5 boundary, the spectrum of (SD_\Omega) satisfies a non resonant property introduced by C. Foias and J. C. Saut to linearize the Navier-Stokes system in a bounded domain \Omega of R^3 with Dirichlet boundary conditions. For that purpose, we first prove that, generically with respect to the domain \Omega with C^5 boundary, all the eigenvalues of (SD_\Omega) are simple. That answers positively a question raised by J. H. Ortega and E. Zuazua. 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.