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 consid- ered. The coupling takes place on the boundary (interface) via the con- tinuity 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 con- trast, in the case of small but rapid oscillations of the interface, in (3 )t he authors established existence of finite energy weak solutions that are de- fined globally. This is achieved by exploiting new hyperbolic trace regu- larity 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.

[1]  Max Gunzburger,et al.  Global Existence of Weak Solutions for Viscous Incompressible Flows around a Moving Rigid Body in Three Dimensions , 2000 .

[2]  L. Hou,et al.  ANALYSIS OF A LINEAR FLUID-STRUCTURE INTERACTION PROBLEM , 2003 .

[3]  M. Horn.  Sharp trace regularity for the solutions of the equations of dynamic elasticity , 1996 .

[4]  Daniel Coutand,et al.  Motion of an Elastic Solid inside an Incompressible Viscous Fluid , 2005 .

[5]  Muriel Boulakia,et al.  Modélisation et analyse mathématique de problèmes d'interaction fluide-structure , 2004 .

[6]  S. Agmon,et al.  Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , 1959 .

[7]  R. Glowinski,et al.  A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow , 2001 .

[8]  Suncica Canic,et al.  Self-Consistent Effective Equations Modeling Blood Flow in Medium-to-Large Compliant Arteries , 2005, Multiscale Model. Simul..

[9]  Irena Lasiecka,et al.  Tangential boundary stabilization of Navier-Stokes equations , 2006 .

[10]  E. Boschi.  Recensioni: J. L. Lions - Quelques méthodes de résolution des problémes aux limites non linéaires. Dunod, Gauthier-Vi;;ars, Paris, 1969; , 1971 .

[11]  Miguel Angel Fernández,et al.  An exact Block–Newton algorithm for solving fluid–structure interaction problems , 2003 .

[12]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[13]  Jean-Paul Zolesio,et al.  Moving Shape Analysis and Control: Applications to Fluid Structure Interactions , 2006 .

[14]  Viorel Barbu,et al.  Flow Invariance Preserving Feedback Controllers for the Navier–Stokes Equation , 2001 .

[15]  G. Truskey,et al.  Three-dimensional numerical simulation of receptor-mediated leukocyte adhesion to surfaces: Effects of cell deformability and viscoelasticity , 2005 .

[16]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[17]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[18]  Paul A Dayton,et al.  The magnitude of radiation force on ultrasound contrast agents. , 2002, The Journal of the Acoustical Society of America.

[19]  Richard B. Melrose,et al.  Singularities of boundary value problems. I , 1978 .

[20]  M. Tucsnak,et al.  Global Weak Solutions¶for the Two-Dimensional Motion¶of Several Rigid Bodies¶in an Incompressible Viscous Fluid , 2002 .

[21]  H. B. Veiga.  On the Existence of Strong Solutions to a Coupled Fluid-Structure Evolution Problem , 2004 .

[22]  Eduard Feireisl.  On the motion of rigid bodies in a viscous incompressible fluid , 2003 .

[23]  J. Lions,et al.  Non homogeneous boundary value problems for second order hyperbolic operators , 1986 .

[24]  Irena Lasiecka,et al.  Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions. II. General boundary data , 1991 .

[25]  Céline Grandmont,et al.  Weak solutions for a fluid-elastic structure interaction model , 2001 .

[26]  Daniel A Hammer,et al.  Effect of microvillus deformability on leukocyte adhesion explored using adhesive dynamics simulations. , 2005, Biophysical journal.

[27]  I. Lasiecka,et al.  The dynamical Lame system: regularity of solutions, boundary controllability and boundary data continuation , 2002 .

[28]  Viorel Barbu,et al.  Differential equations in Banach spaces , 1976 .