Optimal tubes for non-cylindrical Navier-Stokes flows with Navier boundary condition

Abstract

We consider a shape optimization problem where state equations are the time-dependent Navier-Stokes equations with Navier boundary condition occupying a non-cylindrical domain. In the first part of our work, we prove existence of weak solutions for the state equations using the elliptic regularization method. In the second part, we show stability of the state problem solutions with respect to admissible shape variations and derive an existence result for the considered shape minimization problem.