# American Institute of Mathematical Sciences

March  2013, 2(1): 55-79. doi: 10.3934/eect.2013.2.55

## Sensitivity analysis for a free boundary fluid-elasticity interaction

 1 NC State University, Department of Mathematics, 3236 SAS Hall, Raleigh, NC 27695-8205 2 CNRS-INLN, 1361 Routes des Lucioles, Sophia Antipolis, F-06560 Valbonne

Received  June 2012 Revised  September 2012 Published  January 2013

In this paper a total linearization is derived for the free boundary nonlinear elasticity - incompressible fluid interaction. The equations and the free boundary are linearized together and the new linearization turns out to be different from the usual coupling of classical linear models. New extra terms are present on the common interface, some of them involving the boundary curvatures. These terms play an important role in the final linearized system and can not be neglected.
Citation: Lorena Bociu, Jean-Paul Zolésio. Sensitivity analysis for a free boundary fluid-elasticity interaction. Evolution Equations and Control Theory, 2013, 2 (1) : 55-79. doi: 10.3934/eect.2013.2.55
##### References:
 [1] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. Part I: Explicit semigroup generator and its spectral properties, in "Fluids and Waves," Contemp. Math., 440, AMS, Providence, RI, (2007), 15-54. doi: 10.1090/conm/440/08475. [2] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in "Fluids and Waves," Contem. Math., 440, AMS, Providence, RI, (2007), 55-82. doi: 10.1090/conm/440/08476. [3] 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), 1173-1207. doi: 10.1512/iumj.2008.57.3284. [4] L. Bociu, Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping, Nonlinear Analysis A: Theory, Methods and Applications, 71 (2009), e560-e575. doi: 10.1016/j.na.2008.11.062. [5] L. Bociu and I. Lasiecka, Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping, Applicationes Mathematicae, 35 (2008), 281-304. doi: 10.4064/am35-3-3. [6] L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860. doi: 10.3934/dcds.2008.22.835. [7] L. Bociu and I. Lasiecka, Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, JDE, 249 (2010), 654-683. doi: 10.1016/j.jde.2010.03.009. [8] L. Bociu and P. Radu, Existence and uniqueness of weak solutions to the cauchy problem of a semilinear wave equation with supercritical interior source and damping,, Discrete Contin. Dyn. Syst., 2009 (): 60. [9] L. Bociu and J.-P. Zolésio, Linearization of a coupled system of nonlinear elasticity and viscous fluid, in "Modern Aspects of the Theory of Partial Differential Equations," Operator Theory: Advances and Applications, 216, Birkhäuser/Springer Basel AG, Basel, (2011), 93-120. doi: 10.1007/978-3-0348-0069-3_6. [10] L. Bociu and J.-P. Zolésio, Existence for the linearization of a steady state fluid - nonlinear elasticity interaction, Discrete and Continuous Dynamical Systems, Supplement, (2011), 184-197. [11] S. Boisgérault and J. P. Zolésio, Boundary variations in the Navier-Stokes equations and Lagrangian functionals, in "Shape Optimization and Optimal Design" (Cambridge, 1999), Lecture Notes in Pure and Appl. Math., 216, Dekker, New York, (2001), 7-26. [12] M. Boulakia, Existence of weak solutions for the three dimensional motion of an elastic structure in an incompressible fluid, J. Math. Fluid Mech., 9 (2007), 262-294. doi: 10.1007/s00021-005-0201-7. [13] S. Čanić, A. Mikelić, T.-B. Kim and G. Guidoboni, Existence of a unique solution to a nonlinear moving-boundary problem of mixed type arising in modeling blood flow, in "Nonlinear Conservation Laws and Applications," IMA Vol. Math. Appl., 153, Springer, New York, (2011), 235-256. doi: 10.1007/978-1-4419-9554-4_11. [14] P. G. Ciarlet, "Mathematical Elasticity. Volume I: Three-dimensional Elasticity," Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988. [15] C. Conca, J. San Martin and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25 (2000), 1019-1042. doi: 10.1080/03605300008821540. [16] D. Coutand and S. Shkoller, Motion of an elastic solid inside and incompressible viscous fluid, Arch. Rational Mech. Anal., 176 (2005), 25-102. doi: 10.1007/s00205-004-0340-7. [17] D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Rational Mech. Anal. 179 (2006), 303-352. doi: 10.1007/s00205-005-0385-2. [18] M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus and Optimization," Advances in Design and Control, 4, SIAM, Philadelphia, PA, 2001. [19] M. C. Delfour and J.-P. Zolésio, Hidden boundary smoothness for some classes of differential equations on submanifolds, in " Optimization Methods in Partial Differential Equations" (South Hadley, MA, 1996), Contemporary Mathematics, 209, AMS, Providence, RI, (1997), 59-73. doi: 10.1090/conm/209/02759. [20] F. R. Desaint and J.-P. Zolésio, Manifold derivative in the Laplace-Beltrami equation, J. Funct. Anal., 151 (1997), 234-269. doi: 10.1006/jfan.1997.3130. [21] B. Desjardins, M. J. Esteban, C. Grandmont and P. Le Tallec, Weak solutions for a fluid-elastic structure interaction model, Rev. Math. Complut., 14 (2001), 523-538. [22] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71. doi: 10.1007/s002050050136. [23] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, DCDS, 9 (2003), 633-650. doi: 10.3934/dcds.2003.9.633. [24] R. Dziri and J.-P. Zolésio, Dynamical shape control in non-cylindrical Navier-Stokes equations, J. Convex Anal., 6 (1999), 293-318. [25] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equations, 3 (2003), 419-441. doi: 10.1007/s00028-003-0110-1. [26] L. Formaggia, A. Quarteroni and A. Veneziani, eds., "Cardiovascular Mathematics. Modeling and Simulation of the Circulatory System," MS$&$A, Modeling, Simulation and Applications, 1, Springer-Verlag Italia, Milano, 2009. doi: 10.1007/978-88-470-1152-6. [27] C. Grandmont and Y. Maday, Existence for unsteady fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 34 (2000), 609-636. doi: 10.1051/m2an:2000159. [28] M. D. Gunzburger, H.-C. Lee and G. A. Seregin, Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2 (2000), 219-266. doi: 10.1007/PL00000954. [29] E. Kaya, E. Aulisa, A. Ibragimov and P. Seshaiyer, A stability estimate for fluid structure interaction problem with non-linear beam,, DCDS-S, 2009 (): 424. [30] T. Kim, S. Čanić and G. Guidoboni, Existence and uniqueness of a solution to a three-dimensional axially symmetric biot problem arising in modeling blood flow, Communications on Pure and Applied Analysis, 9 (2010), 839-865. doi: 10.3934/cpaa.2010.9.839. [31] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Diff. Eq., 247 (2009), 1452-1478. doi: 10.1016/j.jde.2009.06.005. [32] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary, Nonlinearity, 24 (2011), 159-176. doi: 10.1088/0951-7715/24/1/008. [33] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories," Volumes I and II, Cambridge, University Press, 2000. [34] I.Lasiecka and A. Tuffaha, Optimal feedback synthesis for bolza control problem arising in linearized fluid structure interaction, in " Optimal Control of Coupled Systems of Partial Differential Equations," International Series of Numerical Mathematics, 158, Birkhäuser Verlag, Basel, (2009), 171-190. doi: 10.1007/978-3-7643-8923-9_10. [35] J.-L. Lions, "Quelques Méthodes de Résolution des Problemes aux Limites Non Linéaires," Dunod, 1969. [36] P. I. Plotnikov and J. Sokolowski, Shape derivative of drag functional, SIAM J. Control Optim., 48 (2010), 4680-4706. doi: 10.1137/090758179. [37] J. A. San Martin, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147. doi: 10.1007/s002050100172. [38] J. Sokolowski and J.-P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis," Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9. [39] B. N. Steele, D. Valdez-Jasso, M. A. Haider and M. S. Olufsen, Predicting arterial flow and pressure dynamics using a 1D fluid dynamics model with a viscoelastic wall, SIAM J. Appl. Math., 71 (2011), 1123-1143. doi: 10.1137/100810186. [40] D. Tataru, On the regularity of boundary traces for the wave equation, Annali di Scuola Normale Sup. Pisa Cl. Sci. (4), 26 (1998), 185-206. [41] J.-P. Zolésio, Weak shape formulation of free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21 (1994), 11-44.

show all references

##### References:
 [1] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. Part I: Explicit semigroup generator and its spectral properties, in "Fluids and Waves," Contemp. Math., 440, AMS, Providence, RI, (2007), 15-54. doi: 10.1090/conm/440/08475. [2] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in "Fluids and Waves," Contem. Math., 440, AMS, Providence, RI, (2007), 55-82. doi: 10.1090/conm/440/08476. [3] 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), 1173-1207. doi: 10.1512/iumj.2008.57.3284. [4] L. Bociu, Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping, Nonlinear Analysis A: Theory, Methods and Applications, 71 (2009), e560-e575. doi: 10.1016/j.na.2008.11.062. [5] L. Bociu and I. Lasiecka, Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping, Applicationes Mathematicae, 35 (2008), 281-304. doi: 10.4064/am35-3-3. [6] L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860. doi: 10.3934/dcds.2008.22.835. [7] L. Bociu and I. Lasiecka, Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, JDE, 249 (2010), 654-683. doi: 10.1016/j.jde.2010.03.009. [8] L. Bociu and P. Radu, Existence and uniqueness of weak solutions to the cauchy problem of a semilinear wave equation with supercritical interior source and damping,, Discrete Contin. Dyn. Syst., 2009 (): 60. [9] L. Bociu and J.-P. Zolésio, Linearization of a coupled system of nonlinear elasticity and viscous fluid, in "Modern Aspects of the Theory of Partial Differential Equations," Operator Theory: Advances and Applications, 216, Birkhäuser/Springer Basel AG, Basel, (2011), 93-120. doi: 10.1007/978-3-0348-0069-3_6. [10] L. Bociu and J.-P. Zolésio, Existence for the linearization of a steady state fluid - nonlinear elasticity interaction, Discrete and Continuous Dynamical Systems, Supplement, (2011), 184-197. [11] S. Boisgérault and J. P. Zolésio, Boundary variations in the Navier-Stokes equations and Lagrangian functionals, in "Shape Optimization and Optimal Design" (Cambridge, 1999), Lecture Notes in Pure and Appl. Math., 216, Dekker, New York, (2001), 7-26. [12] M. Boulakia, Existence of weak solutions for the three dimensional motion of an elastic structure in an incompressible fluid, J. Math. Fluid Mech., 9 (2007), 262-294. doi: 10.1007/s00021-005-0201-7. [13] S. Čanić, A. Mikelić, T.-B. Kim and G. Guidoboni, Existence of a unique solution to a nonlinear moving-boundary problem of mixed type arising in modeling blood flow, in "Nonlinear Conservation Laws and Applications," IMA Vol. Math. Appl., 153, Springer, New York, (2011), 235-256. doi: 10.1007/978-1-4419-9554-4_11. [14] P. G. Ciarlet, "Mathematical Elasticity. Volume I: Three-dimensional Elasticity," Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988. [15] C. Conca, J. San Martin and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25 (2000), 1019-1042. doi: 10.1080/03605300008821540. [16] D. Coutand and S. Shkoller, Motion of an elastic solid inside and incompressible viscous fluid, Arch. Rational Mech. Anal., 176 (2005), 25-102. doi: 10.1007/s00205-004-0340-7. [17] D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Rational Mech. Anal. 179 (2006), 303-352. doi: 10.1007/s00205-005-0385-2. [18] M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus and Optimization," Advances in Design and Control, 4, SIAM, Philadelphia, PA, 2001. [19] M. C. Delfour and J.-P. Zolésio, Hidden boundary smoothness for some classes of differential equations on submanifolds, in " Optimization Methods in Partial Differential Equations" (South Hadley, MA, 1996), Contemporary Mathematics, 209, AMS, Providence, RI, (1997), 59-73. doi: 10.1090/conm/209/02759. [20] F. R. Desaint and J.-P. Zolésio, Manifold derivative in the Laplace-Beltrami equation, J. Funct. Anal., 151 (1997), 234-269. doi: 10.1006/jfan.1997.3130. [21] B. Desjardins, M. J. Esteban, C. Grandmont and P. Le Tallec, Weak solutions for a fluid-elastic structure interaction model, Rev. Math. Complut., 14 (2001), 523-538. [22] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71. doi: 10.1007/s002050050136. [23] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, DCDS, 9 (2003), 633-650. doi: 10.3934/dcds.2003.9.633. [24] R. Dziri and J.-P. Zolésio, Dynamical shape control in non-cylindrical Navier-Stokes equations, J. Convex Anal., 6 (1999), 293-318. [25] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equations, 3 (2003), 419-441. doi: 10.1007/s00028-003-0110-1. [26] L. Formaggia, A. Quarteroni and A. Veneziani, eds., "Cardiovascular Mathematics. Modeling and Simulation of the Circulatory System," MS$&$A, Modeling, Simulation and Applications, 1, Springer-Verlag Italia, Milano, 2009. doi: 10.1007/978-88-470-1152-6. [27] C. Grandmont and Y. Maday, Existence for unsteady fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 34 (2000), 609-636. doi: 10.1051/m2an:2000159. [28] M. D. Gunzburger, H.-C. Lee and G. A. Seregin, Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2 (2000), 219-266. doi: 10.1007/PL00000954. [29] E. Kaya, E. Aulisa, A. Ibragimov and P. Seshaiyer, A stability estimate for fluid structure interaction problem with non-linear beam,, DCDS-S, 2009 (): 424. [30] T. Kim, S. Čanić and G. Guidoboni, Existence and uniqueness of a solution to a three-dimensional axially symmetric biot problem arising in modeling blood flow, Communications on Pure and Applied Analysis, 9 (2010), 839-865. doi: 10.3934/cpaa.2010.9.839. [31] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Diff. Eq., 247 (2009), 1452-1478. doi: 10.1016/j.jde.2009.06.005. [32] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary, Nonlinearity, 24 (2011), 159-176. doi: 10.1088/0951-7715/24/1/008. [33] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories," Volumes I and II, Cambridge, University Press, 2000. [34] I.Lasiecka and A. Tuffaha, Optimal feedback synthesis for bolza control problem arising in linearized fluid structure interaction, in " Optimal Control of Coupled Systems of Partial Differential Equations," International Series of Numerical Mathematics, 158, Birkhäuser Verlag, Basel, (2009), 171-190. doi: 10.1007/978-3-7643-8923-9_10. [35] J.-L. Lions, "Quelques Méthodes de Résolution des Problemes aux Limites Non Linéaires," Dunod, 1969. [36] P. I. Plotnikov and J. Sokolowski, Shape derivative of drag functional, SIAM J. Control Optim., 48 (2010), 4680-4706. doi: 10.1137/090758179. [37] J. A. San Martin, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147. doi: 10.1007/s002050100172. [38] J. Sokolowski and J.-P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis," Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9. [39] B. N. Steele, D. Valdez-Jasso, M. A. Haider and M. S. Olufsen, Predicting arterial flow and pressure dynamics using a 1D fluid dynamics model with a viscoelastic wall, SIAM J. Appl. Math., 71 (2011), 1123-1143. doi: 10.1137/100810186. [40] D. Tataru, On the regularity of boundary traces for the wave equation, Annali di Scuola Normale Sup. Pisa Cl. Sci. (4), 26 (1998), 185-206. [41] J.-P. Zolésio, Weak shape formulation of free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21 (1994), 11-44.
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