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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 |
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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