The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow

Abstract

This paper is concerned with the computation of the drag T associated with a body traveling at uniform velocity in a fluid governed by the stationary Navier–Stokes equations. It is assumed that the fluid fills a domain of the form Ω+u, where Ω ⊂ R3 is a reference domain and u is a displacement field. We assume only that Ω is a Lipschitz domain and that u is Lipschitz-continuous. We prove that, at least when the velocity of the body is sufficiently small, u 7→ T(Ω + u) is a C∞ mapping (in a ball centered at 0). We also compute the derivative at 0.