The Differentiability of the Drag with Respect to the Variations of a Lipschitz Domain in a Navier--Stokes Flow
Juan A. Bello, Enrique Fernández‐Cara, J M Lemoine, Jacques Simon · SIAM Journal on Control and Optimization · 1997
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 $\Omega+u$, where $\Omega\subset\reels^3$ is a reference domain and u is a displacement field. We assume only that $\Omega$ is a Lipschitz domain and that u is Lipschitz-continuous. We prove that, at least when the velocity of the body is sufficiently small, $u\mapsto T(\Omega+u)$ is a $C^{\infty}$ mapping (in a ball centered at 0). We also compute the derivative at 0.