Asymptotic Behaviors for Eigenvalues and Eigenfunctions Associated to Stokes Operator in the Presence of Small Boundary Perturbations

Abstract

We consider the Stokes eigenvalue problem in a bounded domain of R 3 with Dirichlet boundary conditions. The aim of this paper is to advance the development of high-order terms in the asymptotic expansions of the boundary perturbations of eigenvalues, eigenfunctions and eigenpressures for the Stokes operator caused by small perturbations of the boundary. Our derivation is rigorous and proved by layer potential techniques.

Small pertubations of an interface for Stokes system

“…We now develop a boundary integral formulation for solving the perturbed Stokes problem (10). The components of the fundamental Stokes tensor Γ = (Γ ) , =1 and those of the associated pressure vector = ( ) , =1 , which determine the fundamental solution (Γ, ) of the Stokes system in ℝ , are given for = 3 by (see for instance [2,19])…”

“…where  Ω ( ) is the hydrodynamic single-layer potential given by (23). The asymptotic expansion of  * is given by the following lemma, see [10,16] for an idea of the proof.…”

“…Proof 3.1. According to [10], the solution of (6) may satisfy the following transmission conditions along the interface :…”

“…To find a representation to solutions of (10) or (6), the reader may see [2,3,6,10,11,20] and others. The solution ( , ) to (10) can be represented by…”

“…In this paper, we suppose that any shape deformation like (7) occurs inside a bounded domain Ω in a Hölder space. Based on the works [1][2][3]6,9,10,12,20], we may advance the asymptotic expansions for the perturbed solutions of the Stokes system and we may give explicitly both the first term correction and a relationship between Stokes solutions measurements and the shape of the object. Since the Stokes system can be viewed as the incompressible limit ( ,̃→ +∞) of the Lamé system, [2,3] it is natural to view the viscosity moment tensor VMT as a limit of the elastic moment tensors EMT.…”