Skip to main content
Log in

Weak solution to the incompressible viscous fluid and a thermoelastic plate interaction problem in 3D

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 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

    Article  MathSciNet  Google Scholar 

  2. Breit D, Schwarzacher S. Compressible fluids interacting with a linear-elastic shell. Arch Ration Mech Anal, 2017, 228: 495–562

    Article  MathSciNet  Google Scholar 

  3. 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

    Article  MathSciNet  Google Scholar 

  4. Chueshov I. Dynamics of a nonlinear elastic plate interacting with a linearized compressible fluid. Nonlinear Anal, 2013, 95: 650–665

    Article  MathSciNet  Google Scholar 

  5. Chueshov I. Interaction of an elastic plate with a linearized inviscid incompressible fluid. Comm Pure Appl Anal, 2014, 13: 1759–1778

    Article  MathSciNet  Google Scholar 

  6. 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

    MATH  Google Scholar 

  7. Chueshov I, Lasiecka I. Von Karman Evolution Equations. New York: Springer, 2010

    Book  Google Scholar 

  8. Chueshov I, Kolbasin S. Long-time dynamics in plate models with strong nonlinear damping. Comm Pure Appl Anal, 2012, 11: 659–674

    Article  MathSciNet  Google Scholar 

  9. Evans L C. Partial Differential Equations. Graduate Studies in Mathematics Vol 19. 2nd ed. Providence, RI: Amer Math Soc, 2010

    Google Scholar 

  10. 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

    Google Scholar 

  11. 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

    Article  MathSciNet  Google Scholar 

  12. Grandmont C, Hillairet M. Existence of global strong solutions to a beam-fluid interaction system. Arch Ration Mech Anal, 2016, 220: 1283–1333

    Article  MathSciNet  Google Scholar 

  13. 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

    Article  MathSciNet  Google Scholar 

  14. 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

    Google Scholar 

  15. Lequeurre J. Existence of strong solutions to a fluid-structure system. SIAM J Math Anal, 2010, 43: 389–410

    Article  MathSciNet  Google Scholar 

  16. 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

    Article  MathSciNet  Google Scholar 

  17. 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

    Article  MathSciNet  Google Scholar 

  18. 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

    Article  MathSciNet  Google Scholar 

  19. Mitra S. Local existence of strong solutions for a fluid-structure interaction model. J Math Fluid Mech, 2020, 22: 60

    Article  MathSciNet  Google Scholar 

  20. 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

    Article  MathSciNet  Google Scholar 

  21. 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

  22. 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

    Article  MathSciNet  Google Scholar 

  23. 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

    MathSciNet  MATH  Google Scholar 

  24. Muha B, Canic S. Existence of a solution to a fluid-multi-layered-structure interaction problem. J Diff Equ, 2014, 256: 658–706

    Article  MathSciNet  Google Scholar 

  25. 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

    Article  MathSciNet  Google Scholar 

  26. 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

    Article  MathSciNet  Google Scholar 

  27. 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

    Article  MathSciNet  Google Scholar 

  28. Ryzhkova I. Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow. Zeitschrift für Angewandte Mathematik und Physik, 2007, 58: 246–261

    Article  MathSciNet  Google Scholar 

  29. 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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Srđan Trifunović.

Additional information

This research was partially supported by National Natural Science Foundation of China (11631008).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-021-0102-8

Key words

2010 MR Subject Classification

Navigation