Advances in Differential Equations

On the interaction problem between a compressible fluid and a Saint-Venant Kirchhoff elastic structure

M. Boulakia and S. Guerrero

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In this paper, we consider an elastic structure immersed in a compressible viscous fluid. The motion of the fluid is described by the compressible Navier-Stokes equations whereas the motion of the structure is given by the nonlinear Saint-Venant Kirchhoff model. For this model, we prove the existence and uniqueness of regular solutions defined locally in time. To do so, we first rewrite the nonlinearity in the elasticity equation in an adequate way. Then, we introduce a linearized problem and prove that this problem admits a unique regular solution. To obtain time regularity on the solution, we use energy estimates on the unknowns and their successive derivatives in time and to obtain spatial regularity, we use elliptic estimates. At last, to come back to the nonlinear problem, we use a fixed point theorem.

Article information

Adv. Differential Equations, Volume 22, Number 1/2 (2017), 1-48.

First available in Project Euclid: 20 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 76N10: Existence, uniqueness, and regularity theory [See also 35L60, 35L65, 35Q30] 74B20: Nonlinear elasticity


Boulakia, M.; Guerrero, S. On the interaction problem between a compressible fluid and a Saint-Venant Kirchhoff elastic structure. Adv. Differential Equations 22 (2017), no. 1/2, 1--48.

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