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

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...

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Bibliographic Details
Published in:arXiv.org 2022-10
Main Author: Simon, John Sebastian H
Format: Article
Language:English
Subjects:
Characteristic functions
Convergence
Domains
Fluid dynamics
Fluid flow
Shape optimization
Time dependence
Topology
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Summary: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.
ISSN:2331-8422