SOLUTIONS TO A FLUID-STRUCTURE INTERACTION FREE BOUNDARY PROBLEM

Our main result is the existence of solutions to the free boundary fluid-structure interaction system. The system consists of a Navier-Stokes equation and a wave equation defined in two different but adjacent domains. The interaction is captured by stress and velocity matching conditions on the free moving boundary lying in between the two domains. We prove the local existence of a solution when the initial velocity of the fluid belongs to $H^{3}$ while the velocity of the elastic body is in $H^{2}$.

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

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

[3]  Roberto Triggiani,et al.  Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface , 2008 .

[4]  R. Kohn,et al.  Partial regularity of suitable weak solutions of the navier‐stokes equations , 1982 .

[5]  Igor Kukavica,et al.  Strong solutions for a fluid structure interaction system , 2010, Advances in Differential Equations.

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

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

[8]  M. Boulakia.  Existence of Weak Solutions for the Three-Dimensional Motion of an Elastic Structure in an Incompressible Fluid , 2007 .

[9]  Igor Kukavica,et al.  Strong solutions to a Navier–Stokes–Lamé system on a domain with a non-flat boundary , 2010 .

[10]  Irena Lasiecka,et al.  Higher Regularity of a Coupled Parabolic-Hyperbolic Fluid-Structure Interactive System , 2008 .

[11]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[12]  Irena Lasiecka,et al.  Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction , 2009, Syst. Control. Lett..

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

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

[15]  M. Boulakia,et al.  A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations , 2009 .

[16]  E. Aulisa,et al.  A stability estimate for fluid structure interaction problem with non-linear beam , 2009 .

[17]  Amjad Tuffaha,et al.  Smoothness of weak solutions to a nonlinear fluid-structure interaction model , 2008 .

[18]  Daniel Coutand,et al.  The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations , 2006 .

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

[20]  Jean-Pierre Raymond,et al.  Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition , 2010 .

[21]  R. Temam.  Navier-Stokes Equations , 1977 .

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

[23]  Luis Vega,et al.  Well-posedness of the initial value problem for the Korteweg-de Vries equation , 1991 .

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

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

[26]  Igor Kukavica,et al.  Strong solutions to a nonlinear fluid structure interaction system , 2009 .

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

[28]  H. B. Veiga.  Navier–Stokes Equations with Shear Thinning Viscosity. Regularity up to the Boundary , 2009 .