A free discontinuity approach to optimal profiles in Stokes flows

TL;DR

This paper develops a free discontinuity approach to identify optimal obstacle shapes in Stokes flows that minimize drag, considering complex geometric features and establishing existence, regularity, and classical solutions in 2D.

Contribution

It introduces a novel free discontinuity framework for shape optimization in Stokes flows, accommodating geometric complexities and proving existence and regularity of solutions.

Findings

Existence of minimal drag obstacle shapes with prescribed volume and surface area.

Regularity results for the optimal shapes, including classical solutions in 2D.

Use of special functions of bounded deformation (SBD) for relaxed solutions.

Abstract

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.