Applied Numerical Mathematics
Shape Derivative of Drag Functional
SIAM Journal on Control and Optimization
Controllability and Time Optimal Control for Low Reynolds Numbers Swimmers
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
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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 $0$.