A free discontinuity approach to optimal profiles in stokes flows

Dorin Bucur (1) , Antonin Chambolle (2, 3) , Alessandro Giacomini (4) , Mickaël Nahon (5)

1 LAMA - Laboratoire de Mathématiques
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
4 Dipartimento di Matematica [Universita di Brescia]
5 MPI-MiS - Max-Planck-Institut für Mathematik in den Naturwissenschaften

In this paper we study obstacles immerged in a Stokes flow with Navier boundary conditions. We prove the existence and regularity of an obstacle with minimal drag, among all shapes of prescribed volume and controlled surface area, taking into account that these shapes may naturally develop geometric features of codimension 1. The existence is carried out in the framework of free discontinuity problems and leads to a relaxed solution in the space of special functions of bounded deformation (SBD). In dimension 2, we prove that the solution is classical.

Keywords: Free discontinuity problems; Stokes flow; Navier boundary conditions; Drag

Domains: Mathematics [math]; Analysis of PDEs [math.AP]; Mathematics [math]; Optimization and Control [math.OC]

Dorin Bucur, Antonin Chambolle, Alessandro Giacomini, Mickaël Nahon. A free discontinuity approach to optimal profiles in stokes flows. 2023. ⟨hal-03920819⟩