Stability of critical shapes for the drag minimization problem in Stokes flow |
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Authors: | Fabien Caubet Marc Dambrine |
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Affiliation: | 1. LMAC, EA 2222, Université de Technologie de Compiègne, F-60200 Compiègne, France;2. LMAP, UMR 5142, Université de Pau et des Pays de l?Adour, CNRS, F-64013 Pau, France |
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Abstract: | 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. |
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Keywords: | 49Q10 49K40 35Q93 76D55 |
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