# 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. Google Scholar [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. Google Scholar [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. Google Scholar [4] G. Avalos and R. Triggiani, Mathematical analysis of PDE systems which govern fluid-structure interactive phenomena,, Bol. Soc. Paran. Mat., 25 (2007), 17. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, "Representation and Control of Infinite Dimensional Systems,", 2nd edition, (2007). Google Scholar [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. Google Scholar [10] F. Bucci, Control-theoretic properties of structural acoustic models with thermal effects, II. Trace regularity results,, Appl. Math., 35 (2008), 305. Google Scholar [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. Google Scholar [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. Google Scholar [13] D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid,, Arch. Ration. Mech. Anal., 176 (2005), 25. Google Scholar [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. Google Scholar [15] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid,, J. Evol. Equ., 3 (2003), 419. Google Scholar [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. Google Scholar [17] I. Lasiecka, "Mathematical Control Theory of Coupled Systems,", CBMS-NSF Regional Conf. Ser. in Appl. Math., 75 (2002). Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [23] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French), Dunod; Gauthier-Villars, (1969). Google Scholar [24] J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Vol. I, (1972). Google Scholar [25] M. Moubachir and J.-P. Zolésio, "Moving Shape Analysis and Control. Applications to Fluid Structure Interactions,", Chapman & Hall/CRC, (2006). Google Scholar [26] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system,, in, XII (2004), 3. Google Scholar

show all references

##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [4] G. Avalos and R. Triggiani, Mathematical analysis of PDE systems which govern fluid-structure interactive phenomena,, Bol. Soc. Paran. Mat., 25 (2007), 17. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, "Representation and Control of Infinite Dimensional Systems,", 2nd edition, (2007). Google Scholar [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. Google Scholar [10] F. Bucci, Control-theoretic properties of structural acoustic models with thermal effects, II. Trace regularity results,, Appl. Math., 35 (2008), 305. Google Scholar [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. Google Scholar [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. Google Scholar [13] D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid,, Arch. Ration. Mech. Anal., 176 (2005), 25. Google Scholar [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. Google Scholar [15] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid,, J. Evol. Equ., 3 (2003), 419. Google Scholar [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. Google Scholar [17] I. Lasiecka, "Mathematical Control Theory of Coupled Systems,", CBMS-NSF Regional Conf. Ser. in Appl. Math., 75 (2002). Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [23] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French), Dunod; Gauthier-Villars, (1969). Google Scholar [24] J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Vol. I, (1972). Google Scholar [25] M. Moubachir and J.-P. Zolésio, "Moving Shape Analysis and Control. Applications to Fluid Structure Interactions,", Chapman & Hall/CRC, (2006). Google Scholar [26] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system,, in, XII (2004), 3. Google Scholar
 [1] Johannes Elschner, George C. Hsiao, Andreas Rathsfeld. An inverse problem for fluid-solid interaction. Inverse Problems & Imaging, 2008, 2 (1) : 83-120. doi: 10.3934/ipi.2008.2.83 [2] Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681 [3] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83 [4] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations & Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59 [5] David Bourne, Howard Elman, John E. Osborn. A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part II: Analysis of Convergence. Communications on Pure & Applied Analysis, 2009, 8 (1) : 143-160. doi: 10.3934/cpaa.2009.8.143 [6] Stuart S. Antman, David Bourne. A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part I: Formulation, Analysis, and Computations. Communications on Pure & Applied Analysis, 2009, 8 (1) : 123-142. doi: 10.3934/cpaa.2009.8.123 [7] T. Tachim Medjo, Louis Tcheugoue Tebou. Robust control problems in fluid flows. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 437-463. doi: 10.3934/dcds.2005.12.437 [8] Bavo Langerock. Optimal control problems with variable endpoints. Conference Publications, 2003, 2003 (Special) : 507-516. doi: 10.3934/proc.2003.2003.507 [9] Alberto Bressan, Yunho Hong. Optimal control problems on stratified domains. Networks & Heterogeneous Media, 2007, 2 (2) : 313-331. doi: 10.3934/nhm.2007.2.313 [10] M'hamed Kesri. Structural stability of optimal control problems. Communications on Pure & Applied Analysis, 2005, 4 (4) : 743-756. doi: 10.3934/cpaa.2005.4.743 [11] Piermarco Cannarsa, Hélène Frankowska, Elsa M. Marchini. On Bolza optimal control problems with constraints. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 629-653. doi: 10.3934/dcdsb.2009.11.629 [12] T. Zolezzi. Extended wellposedness of optimal control problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 547-553. doi: 10.3934/dcds.1995.1.547 [13] Jean-Pierre de la Croix, Magnus Egerstedt. Analyzing human-swarm interactions using control Lyapunov functions and optimal control. Networks & Heterogeneous Media, 2015, 10 (3) : 609-630. doi: 10.3934/nhm.2015.10.609 [14] Peter Monk, Virginia Selgas. An inverse fluid--solid interaction problem. Inverse Problems & Imaging, 2009, 3 (2) : 173-198. doi: 10.3934/ipi.2009.3.173 [15] Stanisław Migórski. A note on optimal control problem for a hemivariational inequality modeling fluid flow. Conference Publications, 2013, 2013 (special) : 545-554. doi: 10.3934/proc.2013.2013.545 [16] Lorena Bociu, Lucas Castle, Kristina Martin, Daniel Toundykov. Optimal control in a free boundary fluid-elasticity interaction. Conference Publications, 2015, 2015 (special) : 122-131. doi: 10.3934/proc.2015.0122 [17] Giulia Cavagnari. Regularity results for a time-optimal control problem in the space of probability measures. Mathematical Control & Related Fields, 2017, 7 (2) : 213-233. doi: 10.3934/mcrf.2017007 [18] M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223 [19] Giuseppe Buttazzo, Lorenzo Freddi. Optimal control problems with weakly converging input operators. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 401-420. doi: 10.3934/dcds.1995.1.401 [20] Piermarco Cannarsa, Cristina Pignotti, Carlo Sinestrari. Semiconcavity for optimal control problems with exit time. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 975-997. doi: 10.3934/dcds.2000.6.975

2018 Impact Factor: 0.545