On an existence theory for a fluid-beam problem encompassing possible contacts
[Existence de solution autorisant d’éventuels contacts pour un problème d’interaction fluide-structure]
Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 933-971.

Dans cet article, nous considérons un système couplé d’équations aux dérivées partielles modélisant l’interaction entre un fluide visqueux incompressible bi-dimensionnel et une poutre élastique mono-dimensionnelle située sur le bord supérieur du domaine fluide. Après avoir construit un cadre fonctionnel de solutions faibles autorisant les configurations où la poutre est en contact avec le fond de la cavité fluide, l’existence de solutions faibles, globale en temps, est démontrée, que des contacts se produisent ou non. La preuve repose sur l’analyse asymptotique d’un système couplé parabolique-parabolique pour lequel un terme de viscosité est ajouté à la structure, et dont on sait qu’il n’autorise pas les contacts. La limite de viscosité évanescente est alors solution de la formulation faible introduite et autorisant le contact.

In this paper we consider a coupled system of pdes modeling the interaction between a two-dimensional incompressible viscous fluid and a one-dimensional elastic beam located on the upper part of the fluid domain boundary. We design a functional framework to define weak solutions in case of contact between the elastic beam and the bottom of the fluid cavity. We then prove that such solutions exist globally in time regardless a possible contact by approximating the beam equation by a damped beam and letting this additional viscosity vanish.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.162
Classification : 76D05,  35D30,  35Q35,  74F10,  76D03
Mots clés : Équations de Navier-Stokes incompressible, interaction fluide-structure, solutions faibles, modélisation du contact
@article{JEP_2021__8__933_0,
     author = {Jean-J\'er\^ome Casanova and C\'eline Grandmont and Matthieu Hillairet},
     title = {On an existence theory for a fluid-beam problem encompassing possible contacts},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {933--971},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.162},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.162/}
}
Jean-Jérôme Casanova; Céline Grandmont; Matthieu Hillairet. On an existence theory for a fluid-beam problem encompassing possible contacts. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 933-971. doi : 10.5802/jep.162. https://jep.centre-mersenne.org/articles/10.5802/jep.162/

[1] R. A. Adams - Sobolev spaces, Pure and Applied Math., vol. 65, Academic Press, New York-London, 1975 | MR 450957 | Zbl 0314.46030

[2] M. Badra & T. Takahashi - “Gevrey regularity for a system coupling the Navier-Stokes system with a beam equation”, SIAM J. Math. Anal. 51 (2019) no. 6, p. 4776-4814 | Article | MR 4039521

[3] H. Beirão da Veiga - “On the existence of strong solutions to a coupled fluid-structure evolution problem”, J. Math. Fluid Mech. 6 (2004) no. 1, p. 21-52 | Article | MR 2027753

[4] 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) no. 3, p. 368-404 | Article | MR 2166981

[5] H. Fujita & N. Sauer - “On existence of weak solutions of the Navier-Stokes equations in regions with moving boundaries”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 17 (1970), p. 403-420 | MR 298258 | Zbl 0206.39702

[6] C. Grandmont - “Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate”, SIAM J. Math. Anal. 40 (2008) no. 2, p. 716-737 | Article | MR 2438783 | Zbl 1158.74016

[7] C. Grandmont & M. Hillairet - “Existence of global strong solutions to a beam-fluid interaction system”, Arch. Rational Mech. Anal. 220 (2016) no. 3, p. 1283-1333 | Article | MR 3466847

[8] C. Grandmont, M. Hillairet & J. Lequeurre - “Existence of local strong solutions to fluid-beam and fluid-rod interaction systems”, Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019) no. 4, p. 1105-1149 | Article | MR 3955112

[9] G. Grubb & V. A. Solonnikov - “Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods”, Math. Scand. 69 (1991) no. 2, p. 217-290 (1992) | Article | MR 1156428

[10] D. Lengeler & M. Ružička - “Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell”, Arch. Rational Mech. Anal. 211 (2014) no. 1, p. 205-255 | Article | MR 3147436

[11] J. Lequeurre - “Existence of strong solutions to a fluid-structure system”, SIAM J. Math. Anal. 43 (2011) no. 1, p. 389-410 | Article | MR 2765696

[12] J. Lequeurre - “Existence of strong solutions for a system coupling the Navier-Stokes equations and a damped wave equation”, J. Math. Fluid Mech. 15 (2013) no. 2, p. 249-271 | Article | MR 3061763

[13] A. Moussa - “Some variants of the classical Aubin-Lions lemma”, J. Evol. Equ. 16 (2016) no. 1, p. 65-93 | Article | MR 3466213

[14] B. Muha & S. Canić - “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. Rational Mech. Anal. 207 (2013) no. 3, p. 919-968 | Article | MR 3017292 | Zbl 1260.35148

[15] B. Muha & S. Schwarzacher - “Existence and regularity for weak solutions for a fluid interacting with a non-linear shell in 3D”, 2019 | arXiv:1906.01962

[16] J. A. San Martín, V. Starovoitov & M. Tucsnak - “Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid”, Arch. Rational Mech. Anal. 161 (2002) no. 2, p. 113-147 | Article | MR 1870954

[17] V. N. Starovoitov - “Nonuniqueness of a solution to the problem on motion of a rigid body in a viscous incompressible fluid”, J. Math. Sci. (New York) 130 (2005) no. 4, p. 4893-4898 | Article