Abstract
In this paper we deal with a nonlinear interaction problem between an incompressible viscous fluid and a nonlinear thermoelastic plate. The nonlinearity in the plate equation corresponds to nonlinear elastic force in various physically relevant semilinear and quasilinear plate models. We prove the existence of a weak solution for this problem by constructing a hybrid approximation scheme that, via operator splitting, decouples the system into two sub-problems, one piece-wise stationary for the fluid and one time-continuous and in a finite basis for the structure. To prove the convergence of the approximate quasilinear elastic force, we develop a compensated compactness method that relies on the maximal monotonicity property of this nonlinear function.
Similar content being viewed by others
References
Beirão da Veiga H. On the existence of strong solutions to a coupled fluid-structure evolution problem. J Math Fluid Mech, 2004, 6: 21–52
Breit D, Schwarzacher S. Compressible fluids interacting with a linear-elastic shell. Arch Ration Mech Anal, 2017, 228: 495–562
Chambolle A, Desjardins B, Esteban M J, Grandmont C. Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. J Math Fluid Mech, 2005, 7: 368–404
Chueshov I. Dynamics of a nonlinear elastic plate interacting with a linearized compressible fluid. Nonlinear Anal, 2013, 95: 650–665
Chueshov I. Interaction of an elastic plate with a linearized inviscid incompressible fluid. Comm Pure Appl Anal, 2014, 13: 1759–1778
Chueshov I, Lasiecka I. Long-time Behavior of Second Order Evolution Equations with Nonlinear Damping. Memoirs of AMS, Vol 195. Providence, RI: Amer Math Soc, 2008
Chueshov I, Lasiecka I. Von Karman Evolution Equations. New York: Springer, 2010
Chueshov I, Kolbasin S. Long-time dynamics in plate models with strong nonlinear damping. Comm Pure Appl Anal, 2012, 11: 659–674
Evans L C. Partial Differential Equations. Graduate Studies in Mathematics Vol 19. 2nd ed. Providence, RI: Amer Math Soc, 2010
Galdi G P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol I. Springer Tracts in Natural Philosophy, Vol 38. New York: Springer-Verlag, 1994
Grandmont C. Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. SIAM J Math Anal, 2007, 40: 716–737
Grandmont C, Hillairet M. Existence of global strong solutions to a beam-fluid interaction system. Arch Ration Mech Anal, 2016, 220: 1283–1333
Grandmont C, Hillairet M, Lequeurre J. Existence of local strong solutions to fluid-beam and fluid-rod interaction systems. Ann Inst H Poincaré Anal Non Linéaire, 2019, 36: 1105–1149
Hasanyan D, Hovakimyan N, Sasane A J, Stepanyan V. Analysis of nonlinear thermoelastic plate equations. Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, 2: 1514–1519
Lequeurre J. Existence of strong solutions to a fluid-structure system. SIAM J Math Anal, 2010, 43: 389–410
Lequeurre J. Existence of strong solutions for a system coupling the Navier Stokes equations and a damped wave equation. J Math Fluid Mech, 2012, 15: 249–271
Lasiecka I, Maad S, Sasane A. Existence and exponential decay of solutions to a quasilinear thermoelastic plate system. Nonlin Diff Equ Appl, 2008, 15: 689–715
Lengeler D, Růžička M. Weak Solutions for an Incompressible Newtonian Fluid Interacting with a Koiter Type Shell. Arch Ration Mech Anal, 2014, 211: 205–255
Mitra S. Local existence of strong solutions for a fluid-structure interaction model. J Math Fluid Mech, 2020, 22: 60
Muha B. A note on the trace Theorem for domains which are locally subgraph of Holder continuous function. Networks Hete Media, 2014, 9: 191–196
Muha B, Schwarzacher S. Existence and regularity for weak solutions for a fluid interacting with a non-linear shell in 3D. Arxiv: https://arxiv.org/abs/1906.01962
Muha B, Canic S. A generalization of the Aubin-Lions-Simon compactness lemma for problems on moving domains. J Diff Equ, 2019, 266: 8370–8418
Muha B, Canic S. A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof. Comm Inform Sys, 2013, 13: 357–397
Muha B, Canic S. Existence of a solution to a fluid-multi-layered-structure interaction problem. J Diff Equ, 2014, 256: 658–706
Muha B, Canic S. Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition. J Diff Equ, 2016, 260: 8550–8589
Muha B, Canic S. Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls. Arch Ration Mech Anal, 2013, 207: 919–968
Muha B, Canic S. Fluid-structure interaction between an incompressible, viscous 3D fluid and an elastic shell with nonlinear Koiter membrane energy. Interf Free Boundaries, 2015, 17: 465–495
Ryzhkova I. Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow. Zeitschrift für Angewandte Mathematik und Physik, 2007, 58: 246–261
Trifunovic S, Wang Y-G. Existence of a weak solution to the fluid-structure interaction problem in 3D. J Diff Equ, 2020, 268: 1495–1531
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by National Natural Science Foundation of China (11631008).
Rights and permissions
About this article
Cite this article
Trifunović, S., Wang, Y. Weak solution to the incompressible viscous fluid and a thermoelastic plate interaction problem in 3D. Acta Math Sci 41, 19–38 (2021). https://doi.org/10.1007/s10473-021-0102-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-021-0102-8
Key words
- fluid-structure interaction
- incompressible viscous fluid
- nonlinear thermoelastic plate
- three space variables
- weak solution