Long-time behavior of solutions to a fluid dynamic shape optimization problem via phase-field method
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911637351432192 |
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| author | Hinze, Michael Kahle, Christian Simon, John Sebastian H. |
| author_facet | Hinze, Michael Kahle, Christian Simon, John Sebastian H. |
| contents | We investigate the long time behavior of solutions to a shape and topology optimization problem with respect to the time-dependent Navier--Stokes equations. The sought topology is represented by a stationary phase-field that represents a smooth indicator function. The fluid equations are approximated by a porous media approach and are time-dependent. In the latter aspect, the considered problem formulation extends earlier work.
We prove that if the time horizon tends to infinity, minima of the time-dependent problem converge towards minima of the corresponding stationary problem. To do so, a convergence rate with respect to the time horizon, of the values of the objective functional, is analytically derived. This allowed us to prove that the solution to the time-dependent problem converges to a phase-field, as the time horizon goes to infinity, which is proven to be a minimizer for the stationary problem. We validate our results by numerical investigation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13293 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Long-time behavior of solutions to a fluid dynamic shape optimization problem via phase-field method Hinze, Michael Kahle, Christian Simon, John Sebastian H. Optimization and Control Analysis of PDEs 35Q93, 35Q30, 76D55, 35R35 We investigate the long time behavior of solutions to a shape and topology optimization problem with respect to the time-dependent Navier--Stokes equations. The sought topology is represented by a stationary phase-field that represents a smooth indicator function. The fluid equations are approximated by a porous media approach and are time-dependent. In the latter aspect, the considered problem formulation extends earlier work. We prove that if the time horizon tends to infinity, minima of the time-dependent problem converge towards minima of the corresponding stationary problem. To do so, a convergence rate with respect to the time horizon, of the values of the objective functional, is analytically derived. This allowed us to prove that the solution to the time-dependent problem converges to a phase-field, as the time horizon goes to infinity, which is proven to be a minimizer for the stationary problem. We validate our results by numerical investigation. |
| title | Long-time behavior of solutions to a fluid dynamic shape optimization problem via phase-field method |
| topic | Optimization and Control Analysis of PDEs 35Q93, 35Q30, 76D55, 35R35 |
| url | https://arxiv.org/abs/2601.13293 |