We study the existence of strong solutions to a 2d fluid-structure system. The fluid is modelled by the incompressible Navier–Stokes equations. The structure represents a steering gear and is described by two parameters corresponding to angles of deformation. Its equations are derived from a virtual work principle. The global domain represents a wind tunnel and imposes mixed boundary conditions to the fluid velocity. Our method reposes on the analysis of the linearized system. Under a compatibility condition on the initial data, we can guarantee local existence in time of strong solutions to the fluid-structure problem.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2019059
Keywords: Strong solutions, fluid-structure interaction, Navier–Stokes equations, deformable structure
@article{M2AN_2020__54_1_301_0,
author = {Delay, Guillaume},
title = {Existence of strong solutions to a fluid-structure system with a structure given by a finite number of parameters},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {301--333},
year = {2020},
publisher = {EDP Sciences},
volume = {54},
number = {1},
doi = {10.1051/m2an/2019059},
mrnumber = {4058209},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2019059/}
}
TY - JOUR AU - Delay, Guillaume TI - Existence of strong solutions to a fluid-structure system with a structure given by a finite number of parameters JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 301 EP - 333 VL - 54 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2019059/ DO - 10.1051/m2an/2019059 LA - en ID - M2AN_2020__54_1_301_0 ER -
%0 Journal Article %A Delay, Guillaume %T Existence of strong solutions to a fluid-structure system with a structure given by a finite number of parameters %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 301-333 %V 54 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2019059/ %R 10.1051/m2an/2019059 %G en %F M2AN_2020__54_1_301_0
Delay, Guillaume. Existence of strong solutions to a fluid-structure system with a structure given by a finite number of parameters. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 1, pp. 301-333. doi: 10.1051/m2an/2019059
[1] , Sobolev spaces. In: Vol. 65 of Pure and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975). | MR | Zbl
[2] , and , Multiscale modeling of the respiratory tract. Math. Models Methods Appl. Sci. 20 (2010) 59–93. | MR | Zbl | DOI
[3] , and , Computational Fluid-Structure Interaction, Methods and Applications. Wiley (2013). | Zbl
[4] , On the existence of strong solutions to a coupled fluid-structure evolution problem. J. Math. Fluid Mech. 6 (2004) 21–52. | MR | Zbl | DOI
[5] , , and , Representation and control of infinite dimensional systems, 2nd edition. In: Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc, Boston, MA (2007). | MR
[6] , Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid. J. Math. Fluid Mech. 9 (2007) 262–294. | MR | Zbl | DOI
[7] and , A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 777–813. | MR | Zbl | Numdam | DOI
[8] and , Regular solutions of a problem coupling a compressible fluid and an elastic structure. J. Math. Pures Appl. 94 (2010) 341–365. | MR | Zbl | DOI
[9] and , On the interaction problem between a compressible fluid and a Saint-Venant Kirchhoff elastic structure. Adv. Differ. Equ. 22 (2017) 1–48. | MR
[10] , and , Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid. Interfaces Free Bound. 14 (2012) 273–306. | MR | Zbl | DOI
[11] , and , Fluid-rigid structure interaction system with Coulomb’s law. SIAM J. Math. Anal. 49 (2017) 4625–4657. | MR | DOI
[12] , , and , Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. J. Math. Fluid Mech. 7 (2005) 368–404. | MR | Zbl | DOI
[13] and , Les inéquations en mécanique et en physique. Dunod, Paris (1972). Travaux et Recherches Mathématiques, No. 21. | MR | Zbl
[14] , An introduction to the mathematical theory of the Navier–Stokes equations, 2nd edition. In: Springer Monographs in Mathematics. Springer, New York (2011). | MR | Zbl
[15] and , Existence for an unsteady fluid–structure interaction problem. M2AN Math. Model. Numer. Anal. 34 (2000) 609–636. | MR | Zbl | Numdam | DOI
[16] , Elliptic problems in nonsmooth domains. In: Vol. 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2011). | MR | Zbl
[17] , Lack of collision between solid bodies in a 2D incompressible viscous flow. Comm. Partial Differ. Equ. 32 (2007) 1345–1371. | MR | Zbl | DOI
[18] , and , Existence of solutions for the equations modeling the motion of rigid bodies in an ideal fluid. J. Funct. Anal. 259 (2010) 2856–2885. | MR | Zbl | DOI
[19] , Existence of strong solutions to a fluid-structure system. SIAM J. Math. Anal. 43 (2011) 389–410. | MR | Zbl | DOI
[20] and , Problèmes aux limites non homogènes et applications. In: Vol. 1 of Travaux et Recherches Mathématiques, No. 17. Dunod, Paris (1968). | Zbl | MR
[21] , and , Single input controllability of a simplified fluid-structure interaction model. ESAIM Control Optim. Calc. Var. 19 (2013) 20–42. | MR | Zbl | Numdam | DOI
[22] , and , Analysis of a system modelling the motion of a piston in a viscous gas. J. Math. Fluid Mech. 19 (2017) 551–579. | MR | DOI
[23] and , Elliptic equations in polyhedral domains. In: Vol. 162 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2010). | MR | Zbl | DOI
[24] , and , Weak solutions for the motion of a self-propelled deformable structure in a viscous incompressible fluid. Acta Appl. Math. 116 (2011) 329–352. | MR | Zbl | DOI
[25] and , Boundary stabilization of the Navier-Stokes equations in the case of mixed boundary conditions. SIAM J. Control Optim. 53 (2015) 3006–3039. | MR | DOI
[26] , Semigroups of linear operators and applications to partial differential equations. In: Vol. 44 of Applied Mathematical Sciences. Springer-Verlag, New York (1983). | MR | Zbl
[27] , Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 921–951. | MR | Zbl | Numdam | DOI
[28] and , A fluid-structure model coupling the Navier-Stokes equations and the Lamé system. J. Math. Pures Appl. 102 (2014) 546–596. | MR | Zbl | DOI
[29] , , and , An initial and boundary value problem modeling of fish-like swimming. Arch. Ration. Mech. Anal. 188 (2008) 429–455. | MR | Zbl | DOI
[30] , and , Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Ration. Mech. Anal. 161 (2002) 113–147. | MR | Zbl | DOI
[31] , The Navier–Stokes equations. In: Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel (2001). | MR | Zbl
[32] , Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain. Adv. Differ. Equ. 8 (2003) 1499–1532. | MR | Zbl
[33] , and , Stabilization of a fluid-rigid body system. J. Differ. Equ. 259 (2015) 6459–6493. | MR | DOI
Cité par Sources :
