Time-periodic solutions for viscous fluids interacting with nonlinear Koiter plates
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917509825822720 |
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| author | Mîndrilă, Claudiu |
| author_facet | Mîndrilă, Claudiu |
| contents | We prove the existence of time-periodic weak solutions for a fluid-structure interaction system coupling the incompressible Navier-Stokes equations in a three-dimensional moving domain with a nonlinear Koiter plate equation on its upper boundary. The lateral boundary is space-periodic, a natural setting for flow in pipes and channels of periodic cross-section driven by a time-periodic pressure gradient, and the fluid satisfies a no-slip coupling condition at the moving interface. The elastic energy of the plate is governed by the nonlinear Koiter model, which yields an $H^2$-coercive operator and accounts for both membrane and bending effects.
To the best of our knowledge, this is the first result on time-periodic weak solutions for a fluid-structure interaction system with a \emph{nonlinear} elastic energy. The main novelty, compared to our earlier works on the linear case -- a linear elastic plate and a linear Koiter shell respectively -- is the replacement of a two-stage fixed-point procedure -- a Leray-Schauder argument at the discrete level followed by a set-valued Kakutani-Glicksberg-Fan argument at the continuous level -- by a \emph{single} Leray-Schauder fixed point applied directly to the fully coupled Galerkin system. This reduction is not merely a simplification: the nonlinearity of the Koiter energy destroys the convexity of the solution map on which Kakutani-Fan relies, making the two-stage approach of~\cite{Claudiu22} unavailable and the single fixed point the only viable strategy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19051 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Time-periodic solutions for viscous fluids interacting with nonlinear Koiter plates Mîndrilă, Claudiu Analysis of PDEs Mathematical Physics 35Q30, 74F10, 76D05, 74B20 We prove the existence of time-periodic weak solutions for a fluid-structure interaction system coupling the incompressible Navier-Stokes equations in a three-dimensional moving domain with a nonlinear Koiter plate equation on its upper boundary. The lateral boundary is space-periodic, a natural setting for flow in pipes and channels of periodic cross-section driven by a time-periodic pressure gradient, and the fluid satisfies a no-slip coupling condition at the moving interface. The elastic energy of the plate is governed by the nonlinear Koiter model, which yields an $H^2$-coercive operator and accounts for both membrane and bending effects. To the best of our knowledge, this is the first result on time-periodic weak solutions for a fluid-structure interaction system with a \emph{nonlinear} elastic energy. The main novelty, compared to our earlier works on the linear case -- a linear elastic plate and a linear Koiter shell respectively -- is the replacement of a two-stage fixed-point procedure -- a Leray-Schauder argument at the discrete level followed by a set-valued Kakutani-Glicksberg-Fan argument at the continuous level -- by a \emph{single} Leray-Schauder fixed point applied directly to the fully coupled Galerkin system. This reduction is not merely a simplification: the nonlinearity of the Koiter energy destroys the convexity of the solution map on which Kakutani-Fan relies, making the two-stage approach of~\cite{Claudiu22} unavailable and the single fixed point the only viable strategy. |
| title | Time-periodic solutions for viscous fluids interacting with nonlinear Koiter plates |
| topic | Analysis of PDEs Mathematical Physics 35Q30, 74F10, 76D05, 74B20 |
| url | https://arxiv.org/abs/2605.19051 |