Existence of local strong solutions to fluid–beam and fluid–rod interaction systems
Grandmont, Céline1, 2 ; Hillairet, Matthieu3 ; Lequeurre, Julien4
1 Inria de Paris, 75012 Paris, France
2 Sorbonne Université, Lab. Jacques-Louis Lions, 75252 Paris, France
3 Institut Montpelliérain Alexander Grothendieck, CNRS, Univ. Montpellier, France
4 Université de Lorraine, CNRS, IECL, Inria, F-57000 Metz, France
Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 1105-1149.
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We study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. Considering three different structure models, we prove existence of a unique local-in-time strong solution, for which there is no gap between the regularity of the initial data and the regularity of the solution enabling to obtain a blow up alternative. In the case of a damped beam this is an alternative proof (and a generalization to non zero initial displacement) of the result that can be found in [20]. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. The key points consist in studying the coupled system without decoupling the fluid from the structure and to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.

DOI : 10.1016/j.anihpc.2018.10.006
Mots-clés : Fluid–structure interaction, Strong solution, Existence and uniqueness
Grandmont, Céline 1, 2 ; Hillairet, Matthieu 3 ; Lequeurre, Julien 4

1 Inria de Paris, 75012 Paris, France
2 Sorbonne Université, Lab. Jacques-Louis Lions, 75252 Paris, France
3 Institut Montpelliérain Alexander Grothendieck, CNRS, Univ. Montpellier, France
4 Université de Lorraine, CNRS, IECL, Inria, F-57000 Metz, France 
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Grandmont, Céline; Hillairet, Matthieu; Lequeurre, Julien. Existence of local strong solutions to fluid–beam and fluid–rod interaction systems. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 1105-1149. doi : 10.1016/j.anihpc.2018.10.006. http://www.numdam.org/articles/10.1016/j.anihpc.2018.10.006/

[1] Beirão da Veiga, H. On the existence of strong solutions to a coupled fluid–structure evolution problem, J. Math. Fluid Mech., Volume 6 (2004) no. 1, pp. 21–52 | DOI | MR | Zbl

[2] Boulakia, M. Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid, J. Math. Fluid Mech., Volume 9 (2007) no. 2, pp. 262–294 | DOI | MR | Zbl

[3] Boulakia, M.; Guerrero, S. Regular solutions of a problem coupling a compressible fluid and an elastic structure, J. Math. Pures Appl., Volume 94 (2010) no. 4, pp. 341–365 | DOI | MR | Zbl

[4] Boulakia, M.; Schwindt, E.L.; Takahashi, T. Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid, Interfaces Free Bound., Volume 14 (2012) no. 3, pp. 273–306 | DOI | MR | Zbl

[5] J.-J. Casanova, Fluid structure system with boundary conditions involving the pressure. ArXiv e-prints, July 2017. | MR

[6] 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., Volume 7 (2005) no. 3, pp. 368–404 | DOI | MR | Zbl

[7] Chen, S.P.; Triggiani, R. Proof of extensions of two conjectures on structural damping for elastic systems, Pac. J. Math., Volume 136 (1989) no. 1, pp. 15–55 | MR | Zbl

[8] Coutand, D.; Shkoller, S. Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., Volume 176 (2005) no. 1, pp. 25–102 | DOI | MR | Zbl

[9] Coutand, D.; Shkoller, S. The interaction between quasilinear elastodynamics and the Navier–Stokes equations, Arch. Ration. Mech. Anal., Volume 179 (2006) no. 3, pp. 303–352 | DOI | MR | Zbl

[10] Desjardins, B.; Esteban, M.J.; Grandmont, C.; Le Tallec, P. Weak solutions for a fluid–elastic structure interaction model, Rev. Mat. Complut., Volume 14 (2001) no. 2, pp. 523–538 | DOI | MR | Zbl

[11] Galdi, G.P.; Kyed, M. Steady flow of a Navier–Stokes liquid past an elastic body, Arch. Ration. Mech. Anal., Volume 194 (2009) no. 3, pp. 849–875 | DOI | MR | Zbl

[12] Grandmont, C. Existence et unicité de solutions d'un problème de couplage fluide–structure bidimensionnel stationnaire, C. R. Acad. Sci. Paris, Sér. I Math., Volume 326 (1998) no. 5, pp. 651–656 | DOI | MR | Zbl

[13] Grandmont, C. Existence for a three-dimensional steady state fluid–structure interaction problem, J. Math. Fluid Mech., Volume 4 (2002) no. 1, pp. 76–94 | DOI | MR | Zbl

[14] Grandmont, C. On an Unsteady Fluid–Beam Interaction Problem, 2004

[15] Grandmont, C. Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., Volume 40 (2008) no. 2, pp. 716–737 | DOI | MR | Zbl

[16] Grandmont, C.; Hillairet, M. Existence of global strong solutions to a beam–fluid interaction system, Arch. Ration. Mech. Anal., Volume 220 (2016) no. 3, pp. 1283–1333 | DOI | MR | Zbl

[17] Guidoboni, G.; Glowinski, R.; Cavallini, N.; Canic, S. Stable loosely-coupled-type algorithm for fluid–structure interaction in blood flow, J. Comput. Phys., Volume 228 (2009) no. 18, pp. 6916–6937 | DOI | MR | Zbl

[18] Kukavica, I.; Tuffaha, A. Well-posedness for the compressible Navier–Stokes–Lamé system with a free interface, Nonlinearity, Volume 25 (2012) no. 11, pp. 3111–3137 | DOI | MR | Zbl

[19] Lengeler, D.; Růžička, M. Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell, Arch. Ration. Mech. Anal., Volume 211 (2014) no. 1, pp. 205–255 | DOI | MR | Zbl

[20] Lequeurre, J. Existence of strong solutions to a fluid–structure system, SIAM J. Math. Anal., Volume 43 (2011) no. 1, pp. 389–410 | DOI | MR | Zbl

[21] Lequeurre, J. Existence of strong solutions for a system coupling the Navier–Stokes equations and a damped wave equation, J. Math. Fluid Mech., Volume 15 (2013) no. 2, pp. 249–271 | DOI | MR | Zbl

[22] Lions, J.-L.; Magenes, E. Non-Homogeneous Boundary Value Problems and Applications, vol. I, Springer-Verlag, New York–Heidelberg, 1972 (Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181) | MR | Zbl

[23] Muha, B.; Canić, 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., Volume 207 (2013) no. 3, pp. 919–968 | DOI | MR | Zbl

[24] Raoult, A. Construction d'un modèle d'évolution de plaques avec terme d'inertie de rotation, Ann. Mat. Pura Appl. (4), Volume 139 (1985), pp. 361–400 | DOI | MR | Zbl

[25] Raymond, J.-P.; Vanninathan, M. A fluid–structure model coupling the Navier–Stokes equations and the Lamé system, J. Math. Pures Appl. (9), Volume 102 (2014) no. 3, pp. 546–596 | MR | Zbl

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