Smoothness of Weak Solutions to a Nonlinear Fluid-structure Interaction Model

The nonlinear fluid-structure interaction coupling the Navier-Stokes equations with a dynamic system of elasticity is considered. The coupling takes place on the boundary (interface) via the continuity of the normal component of the Cauchy stress tensor. Due to a mismatch of parabolic and hyperbolic...

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Bibliographic Details
Published in:Indiana University mathematics journal 2008-01, Vol.57 (3), p.1173-1207
Main Authors: Barbu, Viorel, Grujić, Zoran, Lasiecka, Irena, Tuffaha, Amjad
Format: Article
Language:English
Subjects:
A priori knowledge
Boundary conditions
College mathematics
Commutators
Differential operators
Exact sciences and technology
Fluids
General mathematics
General, history and biography
Mathematical analysis
Mathematics
Navier Stokes equation
Partial differential equations
Sciences and techniques of general use
Solids
Stress tensors
Topological regularity
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Summary:The nonlinear fluid-structure interaction coupling the Navier-Stokes equations with a dynamic system of elasticity is considered. The coupling takes place on the boundary (interface) via the continuity of the normal component of the Cauchy stress tensor. Due to a mismatch of parabolic and hyperbolic regularity, previous results in the literature dealt with either a regularized version of the model, or with very smooth initial conditions leading to local existence only. In contrast, in the case of small but rapid oscillations of the interface, in [3] the authors established existence of finite energy weak solutions that are defined globally. This is achieved by exploiting new hyperbolic trace regularity results which provide a way to deal with the mismatch of parabolic and hyperbolic regularity. The goal of this paper is to establish regularity of weak solutions, for initial data satisfying the appropriate regularity and compatibility conditions imposed on the interface. It is shown that weak solutions equipped with smooth initial data become classical.
ISSN:0022-2518
1943-5258
DOI:10.1512/iumj.2008.57.3284