Long-Time Behaviour of Shape Design Solutions for the Navier--Stokes Equations

TL;DR

This paper studies the long-term behavior of shape optimization solutions for fluid flow governed by Navier--Stokes equations, showing they converge to stationary solutions over time with numerical validation.

Contribution

It establishes the convergence of dynamic shape solutions to stationary solutions in Navier--Stokes problems using $L^ Infty$-topology and extends the analysis to parabolic/elliptic cases.

Findings

Dynamic shape solutions approach stationary solutions over large time horizons.

Convergence is characterized in the $L^ Infty$-topology of characteristic functions.

Numerical examples confirm the theoretical convergence results.

Abstract

We investigate the behavior of dynamic shape design problems for fluid flow at large time horizon. In particular, we shall compare the shape solutions of a dynamic shape optimization problem with that of a stationary problem and show that the solution of the former approaches a neighborhood of that of the latter. The convergence of domains is based on the $L^\infty$-topology of their corresponding characteristic functions which is closed under the set of domains satisfying the cone property. As a consequence, we show that the asymptotic convergence of shape solutions for parabolic/elliptic problems is a particular case of our analysis. Lastly, a numerical example is provided to show the occurrence of the convergence of shape design solutions of time-dependent problems with different values of the terminal time T to a shape design solution of the stationary problem.