Shape optimization for stationary Navier-Stokes equations

This work discusses geometric optimization problems governed by stationary Navier-Stokes equations. Optimal domains are proved to exist under the assumption that the family of admissible domains is bounded and satisfies the Lipschitz condition with a uniform constant, and in the absence of the uniqueness property for the state system. Through the parametrization of the admissible shapes by continuous functions defined on a larger universal domain, the optimization parameter becomes a control, i.e. an element of that family of continuous functions. The approximating extension technique via the penalization of the Navier-Stokes equation enables the approximation of the associated shape optimization problem by an optimal control problem. Results on existence and uniqueness are proved for the approximating problem and a gradient-type algorithm is indicated.