Time-periodic solutions for viscous fluids interacting with nonlinear Koiter plates

Claudiu M\^indril\u{a}

arxiv: 2605.19051 · v1 · pith:BFSWSZ7Bnew · submitted 2026-05-18 · 🧮 math.AP · math-ph· math.MP

Time-periodic solutions for viscous fluids interacting with nonlinear Koiter plates

Claudiu M\^indril\u{a} This is my paper

Pith reviewed 2026-05-20 08:11 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP

keywords time-periodic solutionsfluid-structure interactionnonlinear Koiter plateNavier-Stokes equationsweak solutionsLeray-Schauder fixed pointGalerkin approximationmoving domain

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The pith

Time-periodic weak solutions exist for incompressible Navier-Stokes flow coupled to a nonlinear Koiter plate in a space-periodic channel.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes existence of time-periodic weak solutions for a fluid-structure interaction system that couples the incompressible Navier-Stokes equations in a three-dimensional moving domain to a nonlinear Koiter plate equation on the upper boundary. The setup includes space-periodic lateral boundaries and a time-periodic pressure gradient driving the flow, with no-slip coupling at the moving interface. The nonlinear Koiter energy, which incorporates both membrane and bending effects, is H2-coercive but destroys the convexity properties used in earlier linear-plate arguments. This forces a single Leray-Schauder fixed-point argument applied directly to the fully coupled Galerkin system rather than the two-stage procedure that worked for linear cases.

Core claim

Existence of time-periodic weak solutions is proved for the incompressible Navier-Stokes equations in a three-dimensional moving domain coupled with the nonlinear Koiter plate equation on its upper boundary, via a single Leray-Schauder fixed-point theorem applied to the fully coupled Galerkin system; this replaces the unavailable two-stage Kakutani-Glicksberg-Fan argument because the nonlinear Koiter energy destroys convexity of the solution map.

What carries the argument

Single Leray-Schauder fixed point applied directly to the fully coupled Galerkin system, which handles the loss of convexity induced by the nonlinear Koiter energy.

If this is right

Time-periodic weak solutions exist for the fully nonlinear fluid-plate system under space-periodic lateral boundaries and time-periodic pressure forcing.
The single fixed-point strategy succeeds where the two-stage linear argument fails precisely because of the membrane-plus-bending nonlinearity.
The solutions satisfy the no-slip condition at the moving interface and the weak form of both the fluid and plate equations.
The result covers the first existence theory for nonlinear elastic energy in this class of periodic fluid-structure problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same single-step Galerkin fixed-point reduction may apply to other nonlinear shell or plate models that lose convexity.
Numerical schemes built on the coupled Galerkin system could be used to approximate the periodic states directly.
The periodic-channel setting suggests direct relevance to driven flows in pipes with flexible walls under oscillating pressure.

Load-bearing premise

The nonlinear Koiter energy remains sufficiently coercive in H2 to close the a-priori estimates inside the Leray-Schauder degree argument on the Galerkin level.

What would settle it

An explicit counter-example consisting of a specific time-periodic pressure gradient and domain geometry for which no time-periodic weak solution exists would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.19051 by Claudiu M\^indril\u{a}.

**Figure 1.** Figure 1: The moving domains 2.2 The equations Let us denote, here and throughout the work, the time-space domains by I × Ωη := [ t∈I {t} × Ωη (t). The fluid equations. We assume that at each t ∈ I the domain Ωη (t) is filled with fluid, described below. Let u : I × Ωη 7→ R 3 denote the velocity field of the fluid and p : I × Ωη 7→ R is the associated pressure field associated to u. We assume the fluid to be homogen… view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives the first existence result for time-periodic weak solutions in FSI with nonlinear Koiter energy by switching to a single Leray-Schauder fixed point on the coupled Galerkin system.

read the letter

This paper proves existence of time-periodic weak solutions for incompressible Navier-Stokes in a moving 3D domain coupled to a nonlinear Koiter plate with H^2-coercive energy. The lateral boundaries are space-periodic and the interface uses no-slip, which fits periodic channel flows driven by time-periodic pressure gradients. The central claim is new: it is the first such result with nonlinear elastic energy rather than linear plate or shell models from the authors' earlier papers. They replace the two-stage fixed-point procedure (Leray-Schauder at discrete level plus Kakutani-Glicksberg-Fan at continuous level) with a single Leray-Schauder argument applied directly to the fully coupled Galerkin system. The change is presented as necessary because the nonlinear Koiter energy destroys the convexity of the solution map that the set-valued argument relied on. The H^2-coercivity is used to close the a priori estimates, which is a standard property of the model and helps here. The outline is consistent and the motivation for the method adjustment is stated plainly. The soft spots are in the missing details. The abstract does not show the explicit form of the weak formulation, the precise compactness arguments for the nonlinear terms, or how the moving-domain integrals are controlled after the fixed-point step. Those passages can be delicate in FSI even in the linear case, and the nonlinearity could introduce extra terms that need uniform bounds. Without seeing the estimates it is hard to judge whether the single fixed-point step closes without additional regularity or smallness assumptions. This work is for people already working on mathematical FSI existence results, especially those familiar with the linear Koiter or plate predecessors. A reader who knows the functional-analytic toolkit for periodic Navier-Stokes and plate equations will see exactly where the technical adjustment sits. The thinking is clear and the claim is self-contained, so the paper deserves a serious referee rather than a desk reject.

Referee Report

1 major / 2 minor

Summary. The manuscript proves 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, 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. The proof relies on a Galerkin approximation combined with a single Leray-Schauder fixed-point argument applied directly to the fully coupled system, as the nonlinearity destroys the convexity needed for the two-stage Kakutani-Glicksberg-Fan procedure used in prior linear works.

Significance. If the result holds, this is a significant advancement as the first existence result for time-periodic weak solutions in FSI with nonlinear elastic energy. The single-stage Leray-Schauder strategy is a technical improvement over the authors' earlier linear cases. The H^2-coercivity of the nonlinear Koiter energy is invoked to close the a priori estimates. This contributes to the theory by handling more realistic nonlinear models in periodic channel flows.

major comments (1)

[§4] §4 (Galerkin fixed-point map): the application of Leray-Schauder requires a uniform a priori bound on the solution independent of the Galerkin dimension N; the manuscript should explicitly verify that the H^2-coercivity estimate closes this bound without additional smallness assumptions on the time-periodic pressure gradient.

minor comments (2)

[Introduction] The comparison to the linear case in Claudiu22 could briefly recall why the solution map loses convexity under the nonlinear Koiter energy.
[Preliminaries] Notation for the time-dependent domain and the trace operator at the moving interface should be introduced with a dedicated preliminary subsection.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. We address the single major comment below.

read point-by-point responses

Referee: [§4] §4 (Galerkin fixed-point map): the application of Leray-Schauder requires a uniform a priori bound on the solution independent of the Galerkin dimension N; the manuscript should explicitly verify that the H^2-coercivity estimate closes this bound without additional smallness assumptions on the time-periodic pressure gradient.

Authors: We agree that an explicit verification of the N-independent bound is necessary for the Leray-Schauder argument. The a priori estimates in Section 4 are derived directly from the H^2-coercivity of the nonlinear Koiter energy, which controls all nonlinear coupling terms arising in the Galerkin system. These estimates yield a bound on the solution that is independent of the Galerkin dimension N and does not require any smallness assumption on the time-periodic pressure gradient. To make this verification fully explicit as requested, we will add a clarifying remark immediately after the statement of the a priori estimates in Section 4, confirming that the radius of the ball on which the fixed-point map is considered can be chosen independently of N. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for comparison only; central existence proof is independent

full rationale

The paper references its earlier linear works solely to explain the methodological shift necessitated by the nonlinearity of the Koiter energy, which eliminates the convexity required for the Kakutani-Glicksberg-Fan argument used previously. The core result relies on applying a single Leray-Schauder fixed-point theorem directly to the fully coupled Galerkin system, supported by H^2-coercivity estimates and standard functional analysis tools. This self-citation is not load-bearing for the current proof and does not reduce the derivation to prior results by construction. No patterns such as self-definitional steps, fitted inputs as predictions, or ansatz smuggling are present in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The existence proof rests on standard background results in PDE theory and functional analysis; no free parameters or invented physical entities are introduced.

axioms (1)

standard math Standard properties of Sobolev spaces, weak solutions to Navier-Stokes, and Leray-Schauder fixed-point theorem in Banach spaces
Invoked for Galerkin approximation, a priori estimates, and passage to the limit in the moving domain.

pith-pipeline@v0.9.0 · 5807 in / 1409 out tokens · 43578 ms · 2026-05-20T08:11:59.896931+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the nonlinearity of the Koiter energy destroys the convexity of the solution map... single Leray-Schauder fixed point applied directly to the fully coupled Galerkin system
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

⟨K'(η),η⟩≥2K(η) ... H²-coercive operator

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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[1] [1]

A variational approach to hyperbolic evolutions and fluid-structure interactions.J

Barbora Beneˇ sov´ a, Malte Kampschulte, and Sebastian Schwarzacher. A variational approach to hyperbolic evolutions and fluid-structure interactions.J. Eur. Math. Soc., 2023

work page 2023

[2] [2]

M. E. Bogovskij. The solution of some problems of vector analysis related to the operators div and grad. Tr. Semin. S.L. Soboleva 1, 5-40 (1980)., 1980

work page 1980

[3] [3]

Compressible fluids interacting with a linear-elastic shell.Arch

Dominic Breit and Sebastian Schwarzacher. Compressible fluids interacting with a linear-elastic shell.Arch. Ration. Mech. Anal., 228(2):495–562, 2018

work page 2018

[4] [4]

Existence of time-periodic strong solutions to a fluid-structure system.Discrete Contin

Jean-J´ erˆ ome Casanova. Existence of time-periodic strong solutions to a fluid-structure system.Discrete Contin. Dyn. Syst., 39(6):3291–3313, 2019

work page 2019

[5] [5]

Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate.Journal of Mathematical Fluid Mechanics, 7:368–404, 2005

Antonin Chambolle, Beno¯ ıt Desjardins, Maria J Esteban, and C´ eline Grandmont. Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate.Journal of Mathematical Fluid Mechanics, 7:368–404, 2005

work page 2005

[6] [6]

Esteban, and C´ eline Grandmont

Antonin Chambolle, Benoˆ ıt Desjardins, Maria J. Esteban, and C´ eline Grandmont. Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate.J. Math. Fluid Mech., 7(3):368–404, 2005

work page 2005

[7] [7]

P. G. Ciarlet and A. Roquefort. Justification of a two-dimensional nonlinear shell model of Koiter’s type. Chin. Ann. Math., Ser. B, 22(2):129–144, 2001

work page 2001

[8] [8]

Elsevier, 2000

Philippe G Ciarlet.Theory of shells, volume 3. Elsevier, 2000

work page 2000

[9] [9]

Philippe G. Ciarlet. An introduction to differential geometry with applications to elasticity.J. Elasticity, 78-79(1-3):3–201, 2005

work page 2005

[10] [10]

A nonlinear shell model of koiter’s type.Comptes Rendus

Philippe G Ciarlet and Cristinel Mardare. A nonlinear shell model of koiter’s type.Comptes Rendus. Math´ ematique, 356(2):227–234, 2018

work page 2018

[11] [11]

Evans.Partial differential equations, volume 19 ofGrad

Lawrence C. Evans.Partial differential equations, volume 19 ofGrad. Stud. Math.Providence, RI: American Mathematical Society, 2 edition, 2010

work page 2010

[12] [12]

Galdi.An introduction to the mathematical theory of the Navier-Stokes equations

Giovanni P. Galdi.An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I: Linearized steady problems. New York, NY: Springer-Verlag, 1994

work page 1994

[13] [13]

Giovanni P. Galdi. On time-periodic flow of a viscous liquid past a moving cylinder.Arch. Ration. Mech. Anal., 210(2):451–498, 2013

work page 2013

[14] [14]

Giovanni P. Galdi. Viscous flow past a body translating by time-periodic motion with zero average.Arch. Ration. Mech. Anal., 237(3):1237–1269, 2020

work page 2020

[15] [15]

Galdi and Ana L

Giovanni P. Galdi and Ana L. Silvestre. Existence of time-periodic solutions to the Navier-Stokes equations around a moving body.Pac. J. Math., 223(2):251–267, 2006

work page 2006

[16] [16]

A linear isotropic Cosserat shell model including terms up too(h 5)

Ionel-Dumitrel Ghiba, Mircea Bˆ ırsan, and Patrizio Neff. A linear isotropic Cosserat shell model including terms up too(h 5). Existence and uniqueness.Journal of Elasticity, 154:579–605, 2023

work page 2023

[17] [17]

Nonlinear kirchhoff-love shell models derived from the ciarlet-geymonat energy: modelling and well-posedness, 2026

Ionel-Dumitrel Ghiba, Trung Hieu Giang, and Catalina Ureche. Nonlinear kirchhoff-love shell models derived from the ciarlet-geymonat energy: modelling and well-posedness, 2026

work page 2026

[18] [18]

Ionel-Dumitrel Ghiba and Patrizio Neff. Linear constrained Cosserat-shell models including terms up to o(h5): conditional and unconditional existence and uniqueness.Zeitschrift f¨ ur angewandte Mathematik und Physik, 74(2):47, 2023

work page 2023

[19] [19]

Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate.SIAM J

C´ eline Grandmont. Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate.SIAM J. Math. Anal., 40(2):716–737, 2008

work page 2008

[20] [20]

Gurtin.An introduction to continuum mechanics, volume 158 ofMath

Morton E. Gurtin.An introduction to continuum mechanics, volume 158 ofMath. Sci. Eng.Elsevier, Amsterdam, 1981

work page 1981

[21] [21]

Unrestricted deformations of thin elastic structures interacting with fluids.Journal de Math´ ematiques Pures et Appliqu´ ees, 173:96–148, 2023

Malte Kampschulte, Sebastian Schwarzacher, and Gianmarco Sperone. Unrestricted deformations of thin elastic structures interacting with fluids.Journal de Math´ ematiques Pures et Appliqu´ ees, 173:96–148, 2023. 27

work page 2023

[22] [22]

On time-periodic solutions to an interaction problem between compressible viscous fluids and viscoelastic beams.Nonlinearity, 38(1):015005, nov 2024

Ondˇ rej Kreml, V´ aclav M´ acha,ˇS´ arka Neˇ casov´ a, and Srdan Trifunovi´ c. On time-periodic solutions to an interaction problem between compressible viscous fluids and viscoelastic beams.Nonlinearity, 38(1):015005, nov 2024

work page 2024

[23] [23]

Weak solutions for an incompressible, generalized Newtonian fluid interacting with a linearly elastic Koiter type shell.SIAM J

Daniel Lengeler. Weak solutions for an incompressible, generalized Newtonian fluid interacting with a linearly elastic Koiter type shell.SIAM J. Math. Anal., 46(4):2614–2649, 2014

work page 2014

[24] [24]

Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell.Arch

Daniel Lengeler and Michael R˚ uˇ ziˇ cka. Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell.Arch. Ration. Mech. Anal., 211(1):205–255, 2014

work page 2014

[25] [25]

Time-periodic weak solutions for an incompressible New- tonian fluid interacting with an elastic plate.SIAM J

Claudiu Mˆ ındril˘ a and Sebastian Schwarzacher. Time-periodic weak solutions for an incompressible New- tonian fluid interacting with an elastic plate.SIAM J. Math. Anal., 54(4):4139–4162, 2022

work page 2022

[26] [26]

Time-periodic weak solutions for the interaction of an incompressible fluid with a linear koiter type shell under dynamic pressure boundary conditions.ArXiv preprint, 2023

Claudiu Mˆ ındril˘ a and Sebastian Schwarzacher. Time-periodic weak solutions for the interaction of an incompressible fluid with a linear koiter type shell under dynamic pressure boundary conditions.ArXiv preprint, 2023

work page 2023

[27] [27]

PhD thesis, Charles University, Prague, CZ, 2024

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