Stability of critical shapes for the drag minimization problem in Stokes flow

Fabien Caubet (1) , Marc Dambrine (2)

1 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne
2 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

We study the stability of some critical (or equilibrium) shapes in the minimization problem of the energy dissipated by a fluid (i.e. the drag minimization problem) governed by the Stokes equations. We first compute the shape derivative up to the second order, then provide a sufficient condition for the shape Hessian of the energy functional to be coercive at a critical shape. Under this condition, the existence of such a local strict minimum is then proved using a precise upper bound for the variations of the second order shape derivative of the functional with respect to the coercivity and differentiability norms. Finally, for smooth domains, a lower bound of the variations of the drag is obtained in terms of the measure of the symmetric difference of domains.

Mots clés: stability of critical shape; drag minimization; optimal profiles; shape calculus; shape Hessian; Stokes equations

Domaines: Optimisation et contrôle [math.OC]

Fabien Caubet, Marc Dambrine. Stability of critical shapes for the drag minimization problem in Stokes flow. Journal de Mathématiques Pures et Appliquées, 2013, 100 (3), pp.327--346. ⟨10.1016/j.matpur.2013.01.003⟩. ⟨hal-00780730⟩