American Institute of Mathematical Sciences

June  2011, 4(3): 505-521. doi: 10.3934/dcdss.2011.4.505

Regularity of boundary traces for a fluid-solid interaction model

 1 Università degli Studi di Firenze, Dipartimento di Matematica Applicata, Via S. Marta 3, 50139 Firenze 2 Department of Mathematics, University of Virginia, Charlottesville, VA 22904

Received  May 2009 Revised  December 2009 Published  November 2010

We consider a mathematical model for the interactions of an elastic body fully immersed in a viscous, incompressible fluid. The corresponding composite PDE system comprises a linearized Navier-Stokes system and a dynamic system of elasticity; the coupling takes place on the interface between the two regions occupied by the fluid and the solid, respectively. We specifically study the regularity of boundary traces (on the interface) for the fluid velocity field. The obtained trace regularity theory for the fluid component of the system-of interest in its own right-establishes, in addition, solvability of the associated optimal (quadratic) control problems on a finite time interval, along with well-posedness of the corresponding operator Differential Riccati equations. These results complement the recent advances in the PDE analysis and control of the Stokes-Lamé system.
Citation: Francesca Bucci, Irena Lasiecka. Regularity of boundary traces for a fluid-solid interaction model. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 505-521. doi: 10.3934/dcdss.2011.4.505
References:
 [1] P. Acquistapace, F. Bucci and I. Lasiecka, A trace regularity result for thermoelastic equations with application to optimal boundary control,, J. Math. Anal. Appl., 310 (2005), 262. [2] P. Acquistapace, F. Bucci and I. Lasiecka, Optimal boundary control and Riccati theory for abstract dynamics motivated by hybrid systems of PDEs,, Adv. Differential Equations, 10 (2005), 1389. [3] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, I. Explicit semigroup generator and its spectral properties,, Contemp. Math., 440 (2007), 15. [4] G. Avalos and R. Triggiani, Mathematical analysis of PDE systems which govern fluid-structure interactive phenomena,, Bol. Soc. Paran. Mat., 25 (2007), 17. [5] G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface,, Discrete Contin. Dyn. Syst., 22 (2008), 817. [6] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,, Contemp. Math., 440 (2007), 55. [7] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model,, Indiana Univ. Math. J., 57 (2008), 1173. [8] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, "Representation and Control of Infinite Dimensional Systems,", 2nd edition, (2007). [9] M. Boulakia, Existence of weak solutions for an interaction problem between an elastic structure and a compressible viscous fluid,, J. Math. Pures Appl., 84 (2005), 1515. [10] F. Bucci, Control-theoretic properties of structural acoustic models with thermal effects, II. Trace regularity results,, Appl. Math., 35 (2008), 305. [11] F. Bucci and I. Lasiecka, Singular estimates and Riccati theory for thermoelastic plate models with boundary thermal control,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 11 (2004), 545. [12] F. Bucci and I. Lasiecka, Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions,, Calc. Var. Partial Differential Equations, 37 (2010), 217. [13] D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid,, Arch. Ration. Mech. Anal., 176 (2005), 25. [14] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem,, Discrete Contin. Dyn. Syst., 9 (2003), 633. [15] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid,, J. Evol. Equ., 3 (2003), 419. [16] A. V. Fursikov, M. D. Gunzburger and L. S. Hou, Optimal boundary control for the evolutionary Navier-Stokes system: The three-dimensional case,, SIAM J. Control Optim., 43 (2005), 2191. [17] I. Lasiecka, "Mathematical Control Theory of Coupled Systems,", CBMS-NSF Regional Conf. Ser. in Appl. Math., 75 (2002). [18] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories, II. Abstract Hyperbolic-like Systems over a Finite Time Horizon,", Encyclopedia of Mathematics and its Applications, 75 (2000). [19] I. Lasiecka and R. Triggiani, Optimal control and differential Riccati equations under singular estimates for $e^{At}B$ in the absence of analyticity,, in, (2004), 270. [20] I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by $C_0$ semigroups satisfying a singular estimate,, J. Optim. Theory Appl., 136 (2008), 229. [21] I. Lasiecka and A. Tuffaha, Riccati theory and singular estimates for Bolza control problem arising in linearized fluid structure interactions,, Systems Control Lett., 58 (2009), 499. [22] I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators,, J. Math. Pures Appl., 65 (1986), 149. [23] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French), Dunod; Gauthier-Villars, (1969). [24] J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Vol. I, (1972). [25] M. Moubachir and J.-P. Zolésio, "Moving Shape Analysis and Control. Applications to Fluid Structure Interactions,", Chapman & Hall/CRC, (2006). [26] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system,, in, XII (2004), 3.

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References:
 [1] P. Acquistapace, F. Bucci and I. Lasiecka, A trace regularity result for thermoelastic equations with application to optimal boundary control,, J. Math. Anal. Appl., 310 (2005), 262. [2] P. Acquistapace, F. Bucci and I. Lasiecka, Optimal boundary control and Riccati theory for abstract dynamics motivated by hybrid systems of PDEs,, Adv. Differential Equations, 10 (2005), 1389. [3] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, I. Explicit semigroup generator and its spectral properties,, Contemp. Math., 440 (2007), 15. [4] G. Avalos and R. Triggiani, Mathematical analysis of PDE systems which govern fluid-structure interactive phenomena,, Bol. Soc. Paran. Mat., 25 (2007), 17. [5] G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface,, Discrete Contin. Dyn. Syst., 22 (2008), 817. [6] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,, Contemp. Math., 440 (2007), 55. [7] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model,, Indiana Univ. Math. J., 57 (2008), 1173. [8] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, "Representation and Control of Infinite Dimensional Systems,", 2nd edition, (2007). [9] M. Boulakia, Existence of weak solutions for an interaction problem between an elastic structure and a compressible viscous fluid,, J. Math. Pures Appl., 84 (2005), 1515. [10] F. Bucci, Control-theoretic properties of structural acoustic models with thermal effects, II. Trace regularity results,, Appl. Math., 35 (2008), 305. [11] F. Bucci and I. Lasiecka, Singular estimates and Riccati theory for thermoelastic plate models with boundary thermal control,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 11 (2004), 545. [12] F. Bucci and I. Lasiecka, Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions,, Calc. Var. Partial Differential Equations, 37 (2010), 217. [13] D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid,, Arch. Ration. Mech. Anal., 176 (2005), 25. [14] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem,, Discrete Contin. Dyn. Syst., 9 (2003), 633. [15] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid,, J. Evol. Equ., 3 (2003), 419. [16] A. V. Fursikov, M. D. Gunzburger and L. S. Hou, Optimal boundary control for the evolutionary Navier-Stokes system: The three-dimensional case,, SIAM J. Control Optim., 43 (2005), 2191. [17] I. Lasiecka, "Mathematical Control Theory of Coupled Systems,", CBMS-NSF Regional Conf. Ser. in Appl. Math., 75 (2002). [18] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories, II. Abstract Hyperbolic-like Systems over a Finite Time Horizon,", Encyclopedia of Mathematics and its Applications, 75 (2000). [19] I. Lasiecka and R. Triggiani, Optimal control and differential Riccati equations under singular estimates for $e^{At}B$ in the absence of analyticity,, in, (2004), 270. [20] I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by $C_0$ semigroups satisfying a singular estimate,, J. Optim. Theory Appl., 136 (2008), 229. [21] I. Lasiecka and A. Tuffaha, Riccati theory and singular estimates for Bolza control problem arising in linearized fluid structure interactions,, Systems Control Lett., 58 (2009), 499. [22] I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators,, J. Math. Pures Appl., 65 (1986), 149. [23] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French), Dunod; Gauthier-Villars, (1969). [24] J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Vol. I, (1972). [25] M. Moubachir and J.-P. Zolésio, "Moving Shape Analysis and Control. Applications to Fluid Structure Interactions,", Chapman & Hall/CRC, (2006). [26] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system,, in, XII (2004), 3.
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