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April  2012, 32(4): 1355-1389. doi: 10.3934/dcds.2012.32.1355

## Solutions to a fluid-structure interaction free boundary problem

 1 Department of Mathematics, University of Southern California, Los Angeles, CA 90089 2 Department of Mathematics, The Petroleum Institute, Abu Dhabi, United Arab Emirates

Received  September 2010 Revised  June 2011 Published  October 2011

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 $H^{2}$.
Citation: Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355
##### References:
 [1] G. Avalos, I. Lasiecka and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system, Georgian Math. J., 15 (2008), 403-437. [2] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties, in "Fluids and Waves," 15-54, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007. [3] 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," 55-82, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007. [4] 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. [5] H. Beirão da Veiga, Navier-Stokes equations with shear-thickening viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 11 (2009), 233-257. doi: 10.1007/s00021-008-0257-2. [6] 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. [7] M. Boulakia and S. Guerrero, A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 777-813. [8] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. [9] D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176 (2005), 25-102. doi: 10.1007/s00205-004-0340-7. [10] D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352. doi: 10.1007/s00205-005-0385-2. [11] B. Desjardins, M. J. Esteban, C. Grandmont and P. Le Tallec, Weak solutions for a fluid-elastic structure interaction model, Rev. Mat. Complut., 14 (2001), 523-538. [12] 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), 633-650. doi: 10.3934/dcds.2003.9.633. [13] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, Dedicated to Philippe Bénilan, J. Evol. Equ., 3 (2003), 419-441. doi: 10.1007/s00028-003-0110-1. [14] M. À. Fernández and M. Moubachir, An exact block-Newton algorithm for solving fluid-structure interaction problems, C. R. Math. Acad. Sci. Paris, 336 (2003), 681-686. [15] M. A. Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Systems Estim. Control, 8 (1998), 11 pp. (electronic). [16] E. Kaya, E. Aulisa, A. Ibragimov and P. Seshaiyer, A stability estimate for fluid structure interaction problem with non-linear beam,, in, 2009 (): 424. [17] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0. [18] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Differential Equations, 247 (2009), 1452-1478. doi: 10.1016/j.jde.2009.06.005. [19] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system, Adv. Differential Equations, 15 (2010), 231-254. [20] 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. [21] I. Kukavica and A. Tuffaha, Local existence of strong solutions for a free-boundary fluid-structure interaction system,, in preparation., (). [22] J.-L. Lions, "Quelques Méthodes de Résolution des Oroblèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969. [23] I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. (9), 65 (1986), 149-192. [24] J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications," Vol. II, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182, Springer-Verlag, New York-Heidelberg, 1972. [25] I. Lasiecka and A. Tuffaha, Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction, Systems Control Lett., 58 (2009), 499-509. doi: 10.1016/j.sysconle.2009.02.010. [26] M. Moubachir and J.-P. Zolésio, "Moving Shape Analysis and Control. Applications to Fluid Structure Interactions," Pure and Applied Mathematics (Boca Raton), 277, Chapman & Hall/CRC, Boca Raton, FL, 2006. [27] J.-P. Raymond, Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1537-1564. doi: 10.3934/dcdsb.2010.14.1537. [28] J. A. San Martín, 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. [29] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. [30] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.

show all references

##### References:
 [1] G. Avalos, I. Lasiecka and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system, Georgian Math. J., 15 (2008), 403-437. [2] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties, in "Fluids and Waves," 15-54, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007. [3] 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," 55-82, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007. [4] 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. [5] H. Beirão da Veiga, Navier-Stokes equations with shear-thickening viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 11 (2009), 233-257. doi: 10.1007/s00021-008-0257-2. [6] 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. [7] M. Boulakia and S. Guerrero, A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 777-813. [8] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. [9] D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176 (2005), 25-102. doi: 10.1007/s00205-004-0340-7. [10] D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352. doi: 10.1007/s00205-005-0385-2. [11] B. Desjardins, M. J. Esteban, C. Grandmont and P. Le Tallec, Weak solutions for a fluid-elastic structure interaction model, Rev. Mat. Complut., 14 (2001), 523-538. [12] 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), 633-650. doi: 10.3934/dcds.2003.9.633. [13] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, Dedicated to Philippe Bénilan, J. Evol. Equ., 3 (2003), 419-441. doi: 10.1007/s00028-003-0110-1. [14] M. À. Fernández and M. Moubachir, An exact block-Newton algorithm for solving fluid-structure interaction problems, C. R. Math. Acad. Sci. Paris, 336 (2003), 681-686. [15] M. A. Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Systems Estim. Control, 8 (1998), 11 pp. (electronic). [16] E. Kaya, E. Aulisa, A. Ibragimov and P. Seshaiyer, A stability estimate for fluid structure interaction problem with non-linear beam,, in, 2009 (): 424. [17] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0. [18] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Differential Equations, 247 (2009), 1452-1478. doi: 10.1016/j.jde.2009.06.005. [19] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system, Adv. Differential Equations, 15 (2010), 231-254. [20] 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. [21] I. Kukavica and A. Tuffaha, Local existence of strong solutions for a free-boundary fluid-structure interaction system,, in preparation., (). [22] J.-L. Lions, "Quelques Méthodes de Résolution des Oroblèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969. [23] I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. (9), 65 (1986), 149-192. [24] J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications," Vol. II, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182, Springer-Verlag, New York-Heidelberg, 1972. [25] I. Lasiecka and A. Tuffaha, Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction, Systems Control Lett., 58 (2009), 499-509. doi: 10.1016/j.sysconle.2009.02.010. [26] M. Moubachir and J.-P. Zolésio, "Moving Shape Analysis and Control. Applications to Fluid Structure Interactions," Pure and Applied Mathematics (Boca Raton), 277, Chapman & Hall/CRC, Boca Raton, FL, 2006. [27] J.-P. Raymond, Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1537-1564. doi: 10.3934/dcdsb.2010.14.1537. [28] J. A. San Martín, 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. [29] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. [30] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.
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