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

Sensitivity analysis for a free boundary fluid-elasticity interaction

Lorena Bociu 1, and  Jean-Paul Zolésio 2,

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.
Keywords: Navier-Stokes, Free boundary, linearization, coupled system, nonlinear elasticity, shape derivative..
Mathematics Subject Classification: Primary: 93B35, 93B18, 74F10, 35R35; Secondary: 74B20, 35Q3.
Citation: Lorena Bociu, Jean-Paul Zolésio. Sensitivity analysis for a free boundary fluid-elasticity interaction. Evolution Equations & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

P. G. Ciarlet, "Mathematical Elasticity. Volume I: Three-dimensional Elasticity," Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. Dziri and J.-P. Zolésio, Dynamical shape control in non-cylindrical Navier-Stokes equations, J. Convex Anal., 6 (1999), 293-318.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories," Volumes I and II, Cambridge, University Press, 2000. Google Scholar

[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.  Google Scholar

[35]

J.-L. Lions, "Quelques Méthodes de Résolution des Problemes aux Limites Non Linéaires," Dunod, 1969. Google Scholar

[36]

P. I. Plotnikov and J. Sokolowski, Shape derivative of drag functional, SIAM J. Control Optim., 48 (2010), 4680-4706. doi: 10.1137/090758179.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

J.-P. Zolésio, Weak shape formulation of free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21 (1994), 11-44.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

P. G. Ciarlet, "Mathematical Elasticity. Volume I: Three-dimensional Elasticity," Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. Dziri and J.-P. Zolésio, Dynamical shape control in non-cylindrical Navier-Stokes equations, J. Convex Anal., 6 (1999), 293-318.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories," Volumes I and II, Cambridge, University Press, 2000. Google Scholar

[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.  Google Scholar

[35]

J.-L. Lions, "Quelques Méthodes de Résolution des Problemes aux Limites Non Linéaires," Dunod, 1969. Google Scholar

[36]

P. I. Plotnikov and J. Sokolowski, Shape derivative of drag functional, SIAM J. Control Optim., 48 (2010), 4680-4706. doi: 10.1137/090758179.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

J.-P. Zolésio, Weak shape formulation of free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21 (1994), 11-44.  Google Scholar

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