Well-posedness for a 2D/3D fluid-shell interaction model

Wantang Li; Alain Miranville; Xin-Guang Yang; Jinyun Yuan; Wantang Li; Alain Miranville; Xin-Guang Yang; Jinyun Yuan

doi:10.3934/era.2025312

Electronic Research Archive

2025, Volume 33, Issue 11: 7065-7084. doi: 10.3934/era.2025312

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Research article

Well-posedness for a 2D/3D fluid-shell interaction model

1.
Department of Mathematics and Statistics, Henan Normal University, Xinxiang 453007, China
2.
Université Le Havre Normandie, Laboratoire de Mathématiques Appliqués du Havre (LMAH), 25, Rue Philippe Lebon, Le Havre CEDEX 76063, France
3.
Department of Computer Science and Technology, Dongguan University of Technology, Dongguan, China

Received: 11 September 2025 Revised: 09 November 2025 Accepted: 17 November 2025 Published: 21 November 2025

The 2D/3D interaction systems consisting of the incompressible Navier-Stokes equations in a bounded domain and the classical full von Kármán shallow shell equation were investigated in this paper, where the fluid and the shell were coupled via both the transverse and longitudinal displacements. The global existence of weak solutions in 2D/3D was proved by using a special choice of basis functions for the related elliptic equations, which allows one to tackle the delicate estimates for the interaction between the fluid and the shell. Moreover, the uniqueness of the weak solution in 2D was also established via a localization technique.
- fluid-structure interaction model,
- von Kármán equation,
- well-posedness
Citation: Wantang Li, Alain Miranville, Xin-Guang Yang, Jinyun Yuan. Well-posedness for a 2D/3D fluid-shell interaction model[J]. Electronic Research Archive, 2025, 33(11): 7065-7084. doi: 10.3934/era.2025312

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Abstract

The 2D/3D interaction systems consisting of the incompressible Navier-Stokes equations in a bounded domain and the classical full von Kármán shallow shell equation were investigated in this paper, where the fluid and the shell were coupled via both the transverse and longitudinal displacements. The global existence of weak solutions in 2D/3D was proved by using a special choice of basis functions for the related elliptic equations, which allows one to tackle the delicate estimates for the interaction between the fluid and the shell. Moreover, the uniqueness of the weak solution in 2D was also established via a localization technique.

References

[1]	C. H. A. Cheng, D. Coutand, S. Shkoller, Navier-Stokes equations interacting with a nonlinear elastic biofluid shell, SIAM J. Math. Anal., 39 (2007), 742–800. https://doi.org/10.1137/060656085 doi: 10.1137/060656085
[2]	D. Coutand, S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176 (2005), 25–102. https://doi.org/10.1007/s00205-004-0340-7 doi: 10.1007/s00205-004-0340-7
[3]	D. Coutand, S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303–352. https://doi.org/10.1007/s00205-005-0385-2 doi: 10.1007/s00205-005-0385-2
[4]	I. Chueshov, I. Ryzhkova, A global attractor for a fluid-plate interaction model, Commun. Pure Appl. Anal., 12 (2013), 1635–1656. https://doi.org/10.3934/cpaa.2013.12.1635 doi: 10.3934/cpaa.2013.12.1635
[5]	I. Chueshov, I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, J. Differ. Equations, 254 (2013), 1833–1862. https://doi.org/10.1016/j.jde.2012.11.006 doi: 10.1016/j.jde.2012.11.006
[6]	B. Desjardins, M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59–71. https://doi.org/10.1007/s002050050136 doi: 10.1007/s002050050136
[7]	B. Desjardins, M. J. Esteban, C. Grandmont, P. Le Tallec, Weak solutions for a fluid-elastic structure interaction model, Rev. Matemática Complutense, 14 (2001), 523–538. https://doi.org/10.5209/rev_REMA.2001.v14.n2.17030 doi: 10.5209/rev_REMA.2001.v14.n2.17030
[8]	S. Čanić, B. Muha, M. Bukač, Fluid-structure interaction in hemodynamics: modeling, analysis, and numerical simulation, in Fluid-Structure Interaction and Biomedical Applications, Birkhäuser/Springer, Basel, (2014), 79–195.
[9]	B. Kaltenbacher, I, Kuakavica, I. Lasiecka, R. Triggiani, A. Tuffaha, J. T. Webster, Mathematical Theory of Evolutionary Fluid-Flow Structure Interactions, Birkhäuser/Springer, Cham, 2018. https://doi.org/10.1007/978-3-319-92783-1
[10]	D. Lengeler, M. Růžička, Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell, Arch. Ration. Mech. Anal., 211 (2014), 205–255. https://doi.org/10.1007/s00205-013-0686-9 doi: 10.1007/s00205-013-0686-9
[11]	S. Wang, L. Shen, S. Jiang, Modelling and mathematical theory on fluid structure interaction models in aircraft engine, Sci. Sin. Math., 55 (2025), 753–778. https://doi.org/10.1360/SSM-2024-0028 doi: 10.1360/SSM-2024-0028
[12]	B. Muha, S. Čanić, Fluid-structure interaction between an incompressible, viscous 3D fluid and an elastic shell with nonlinear Koiter membrane energy, Interfaces Free Bound., 17 (2015), 465–495. https://doi.org/10.4171/IFB/350 doi: 10.4171/IFB/350
[13]	S. Čanić, Moving boundary problems, Bull. Amer. Math. Soc. (N.S.), 58 (2021), 79–106. https://doi.org/10.1090/bull/1703
[14]	A. Benabdallah, I. Lasiecka, Exponential decay rates for a full von Kármán system of dynamic thermoelasticity, J. Differ. Equations, 160 (2000), 51–93. https://doi.org/10.1006/jdeq.1999.3656 doi: 10.1006/jdeq.1999.3656
[15]	H. Koch, I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, Evolution Equations, Semigroups and Functional Analysis: In Memory of Brunello Terreni. Basel: Birkh?user Basel, (2002), 197–216.
[16]	I. Lasiecka, Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998), 1376–1422. https://doi.org/10.1137/S0363012996301907 doi: 10.1137/S0363012996301907
[17]	J. P. Puel, M. Tucsnak, Global existence for the full von Kármán system, Appl. Math. Optim., 34 (1996), 139–160. https://doi.org/10.4171/IFB/350 doi: 10.4171/IFB/350
[18]	V. I. Sedenko, Classical solvability of an initial-boundary value problem in the nonlinear theory of oscillations of shallow shells, Dokl. Akad. Nauk, 331 (1993), 283–285.
[19]	V. I. Sedenko, On the uniqueness theorem for generalized solutions of initial-boundary problems for the Marguerre-Vlasov vibrations of shallow shells with clamped boundary conditions, Appl. Math. Optim., 39 (1999), 309–326. https://doi.org/10.1007/s002459900108 doi: 10.1007/s002459900108
[20]	I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Math. Methods Appl. Sci., 34 (2011), 1801–1812. https://doi.org/10.1002/mma.1496 doi: 10.1002/mma.1496
[21]	M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech., 10 (2008), 388–401. https://doi.org/10.1007/s00021-006-0236-4 doi: 10.1007/s00021-006-0236-4
[22]	W. Li, P. Piccione, X. G. Yang, J.Y. Yuan, Well-posedness and stability for a fluid-plate interaction model, 2025, submitted.
[23]	Q. Du, M. D. Gunzburger, L. S. Hou, J. Lee, Analysis of a linear fluid-structure interaction problem, Discrete Contin. Dyn. Syst., 9 (2003), 633–650. https://doi.org/10.3934/dcds.2003.9.633 doi: 10.3934/dcds.2003.9.633
[24]	A. Chambolle, B. Desjardins, M. J. Esteban, C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005), 368–404. https://doi.org/10.1007/s00021-004-0121-y doi: 10.1007/s00021-004-0121-y
[25]	V. Barbu, Z. Grujić, I. Lasiecka, A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Contemp. Math., 440 (2007), 55–82. https://doi.org/10.1090/conm/440/08476 doi: 10.1090/conm/440/08476
[26]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969.
[27]	J. C. Robinson, J. L. Rodrigo, W. Sadowski, The Three-Dimensional Navier–Stokes Equations: Classical Theory, Cambridge University Press, 2016. https://doi.org/10.1017/CBO9781139095143
[28]	R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Society, 2024.
[29]	W. Shi, X. G. Yang, L. Shen, Well-posedness for incompressible fluid-solid interaction with vorticity, Commun. Nonlinear Sci. Numer. Simul., 119 (2023), 107113. https://doi.org/10.1016/j.cnsns.2023.107113 doi: 10.1016/j.cnsns.2023.107113

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