Asymptotic formula for the solution of the Stokes problem with a small perturbation of the domain in two and three dimensions

TL;DR

This paper derives an asymptotic formula for the Stokes problem's solution when the domain undergoes a small boundary perturbation, using layer potential techniques to analyze the solution's continuity and displacement field changes.

Contribution

It introduces a rigorous method to obtain the first-order perturbation term for the Stokes problem with potential for higher-order analysis.

Findings

Solution is continuous under small domain perturbations

First-order displacement field term derived

Method allows for higher-order term derivation

Abstract

In this paper we consider the resolvent Stokes problem in the case there is a small perturbation of the domain caused by a perturbed boundary. Firstly, we prove that the solution of Stokes problem is continuous due to this small perturbation. Secondly, we derive the first-order term in the displacement field perturbation that due to the deformation of the domain. It is worth emphasizing that even though only the first-order term is given, our method enables us to derive higher-order terms as well. The derivation is rigorous and based on layer potential techniques.