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Preface: Introduction to the Special Volume on Nonlinear PDEs and Control Theory with Applications
Stability analysis of non-linear plates coupled with Darcy flows
1. | Department of Mathematics and Statistics, Texas Tech University, Lubbock TX, 79409-1042 |
2. | Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States |
References:
[1] |
E. Aulisa, L. Bloshanskaya and A. Ibragimov, Long-term dynamics for well productivity index for nonlinear flows in porous media, Journal of Mathematical Physics, 52 (2011), 023506, 26 pp.
doi: 10.1063/1.3536463. |
[2] |
E. Aulisa, A. Cervone, S. Manservisi and P. Seshaiyer, A multilevel domain decomposition approach for studying coupled flow application, Communications in Computational Physics, 6 (2009), 319-341.
doi: 10.4208/cicp.2009.v6.p319. |
[3] |
Annalisa Quaini, Suncica Canic, Roland Glowinski, Stephen Igo, Craig J Hartley, William Zoghbi and Stephen Little, Validation of a 3D computational fluid-structure interaction model simulating flow through an elastic aperture, Journal of Biomechanics, 45 (2012), 310-318. |
[4] |
I. D. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, Journal of Differential Equations, 233 (2007), 42-86.
doi: 10.1016/j.jde.2006.09.019. |
[5] |
I. D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of the AMS, (2008). |
[6] |
I. D. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Mathematical Methods in the Applied Sciences, 34 (2011), 1801-1812.
doi: 10.1002/mma.1496. |
[7] |
Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, Discrete and Continuous Dynamical Systems, 9 (2003), 633-650.
doi: 10.3934/dcds.2003.9.633. |
[8] |
L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, AMS, Providence, RI, 1998. |
[9] |
F. Flori and P. Orenga, Fluid-structure interaction: Analysis of a 3-D compressible model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 753-777.
doi: 10.1016/S0294-1449(00)00119-0. |
[10] |
D. G. Gorman, I. Trendafilova, A. J. Mulholland and J. Horacek, Analytical modeling and extraction of the modal behavior of a cantilever beam in fluid interaction, Journal of Sound and Vibration, 308 (2007), 231-245. |
[11] |
C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM Journal on Mathematical Analysis, 40 (2008), 716-737.
doi: 10.1137/070699196. |
[12] |
M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid, Journal of Mathematical Fluid Mechanics, 10 (2008), 388-401.
doi: 10.1007/s00021-006-0236-4. |
[13] |
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.
doi: 10.1016/j.anihpc.2008.02.004. |
[14] |
M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure interaction model, Math. Meth. Appl. Sci., 32 (2009), 1452-1466.
doi: 10.1002/mma.1104. |
[15] |
L. Hoang and A. Ibragimov, Structural stability of generalized Forchheimer equations for compressible fluids in porous media, Nonlinearity, 24 (2011), 1-41.
doi: 10.1088/0951-7715/24/1/001. |
[16] |
J. Hron and S. Turek, A monolithic FEM/multigrid solver for ALE formulation of fluid-structure interaction with application in biomechanics, in "Fluid-structure interaction," Lecture Notes in Computational Science and Engineering, 53, Springer, Berlin, (2006), 146-170.
doi: 10.1007/3-540-34596-5_7. |
[17] |
E. Kaya, E. Aulisa, A. Ibragimov and P. Seshaiyer, A stability estimate for fluid structure interaction problem with non-linear beam, Discrete And Continuous Dynamical Systems, Supplement (2009), 424-432. |
[18] |
E. Kaya-Cekin, E. Aulisa, A. Ibragimov and P. Seshaiyer, Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam, Communications on Pure and Applied Analysis, 10 (2011), 1447-1462.
doi: 10.3934/cpaa.2011.10.1447. |
[19] |
E. Kaya-Cekin, E. Aulisa, A. Ibragimov and P. Seshaiyer, Fluid structure interaction problem with changing thickness non-linear beam, Discrete And Continuous Dynamical Systems, Supplement (2011), 813-823. |
[20] |
A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, Journal of Mathematical Analysis and Applications, 318 (2006), 92-101.
doi: 10.1016/j.jmaa.2005.05.031. |
[21] |
H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, in "Evolution Equations, Semigroups and Functional Analysis" (Milano, 2000), Progress in Nonlinear Differential Equations and their Applications, 50, Birkhäuser, Basel, (2002), 197-216. |
[22] |
I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem, Discrete And Continuous Dynamical Systems, 32 (2012), 1355-1389.
doi: 10.3934/dcds.2012.32.1355. |
[23] |
J. E. Lagnese, Modelling and stabilization of nonlinear plates, in "Estimation and Control of Distributed Parameter Systems" (Vorau, 1990), International Series of Numerical Mathematics, 100, Birkhäuser, Basel, (1991), 247-264. |
[24] |
J. E. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Studies in Applied Mathematics, 10, SIAM, Philadelphia, PA, 1989.
doi: 10.1137/1.9781611970821. |
[25] |
I. Lasiecka, Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998), 1376-1422.
doi: 10.1137/S0363012996301907. |
[26] |
V. G. Maz'ya, "Sobolev Spaces," $2^{nd}$ augmented edition, Grundlehren der mathematischen Wissenschaften, Vol. 342, Springer, 2011. |
[27] |
M. Muskat, "The Flow of Homogeneous Fluids Through Porous Media," McGraw-Hill, New York, 1937. |
[28] |
V. V. Novozhilov, "Foundations of the Nonlinear Theory of Elasticity," Dover Publication, INC, 1999. |
[29] |
J. Y. Park and J. R. Kang, Global existence and stability for a von Karman equations with memory in noncylindrical domains, Journal of Mathematical Physics, 50 (2009), 112701, 13 pp.
doi: 10.1063/1.3253977. |
[30] |
J.-P. Puel and M. Tucsnak, Boundary stabilization for the von Kármán equations, SIAM J. Control Optim., 33 (1995), 255-273.
doi: 10.1137/S0363012992228350. |
[31] |
N. Peake and S. V. Sorokin, A nonlinear model of the dynamics of a large elastic plate with heavy fluid loading, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2205-2224.
doi: 10.1098/rspa.2006.1673. |
[32] |
J. Peradze, A numerical algorithm for Kirchhoff-type nonlinear static beam, Journal of Applied Mathematics, 2009 (2009), Art. ID 818269, 12 pp.
doi: 10.1155/2009/818269. |
[33] |
M. Sathyamoorthy, "Nonlinear Analysis of Structures," CRC, 1998. |
[34] |
D. Tataru and M. Tucsnak, On the Cauchy problem for the full von Kármán system, Nonlinear Differential Equations Appl., 4 (1997), 325-340.
doi: 10.1007/s000300050018. |
[35] |
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. |
[36] |
L. Yang and C. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Analysis: Theory, Methods & Applications, 69 (2008), 3802-3810.
doi: 10.1016/j.na.2007.10.016. |
[37] |
SIAM PDE Conference 2011 San-Diego, Book of Abstracts,, Available from: , ().
|
show all references
References:
[1] |
E. Aulisa, L. Bloshanskaya and A. Ibragimov, Long-term dynamics for well productivity index for nonlinear flows in porous media, Journal of Mathematical Physics, 52 (2011), 023506, 26 pp.
doi: 10.1063/1.3536463. |
[2] |
E. Aulisa, A. Cervone, S. Manservisi and P. Seshaiyer, A multilevel domain decomposition approach for studying coupled flow application, Communications in Computational Physics, 6 (2009), 319-341.
doi: 10.4208/cicp.2009.v6.p319. |
[3] |
Annalisa Quaini, Suncica Canic, Roland Glowinski, Stephen Igo, Craig J Hartley, William Zoghbi and Stephen Little, Validation of a 3D computational fluid-structure interaction model simulating flow through an elastic aperture, Journal of Biomechanics, 45 (2012), 310-318. |
[4] |
I. D. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, Journal of Differential Equations, 233 (2007), 42-86.
doi: 10.1016/j.jde.2006.09.019. |
[5] |
I. D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of the AMS, (2008). |
[6] |
I. D. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Mathematical Methods in the Applied Sciences, 34 (2011), 1801-1812.
doi: 10.1002/mma.1496. |
[7] |
Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, Discrete and Continuous Dynamical Systems, 9 (2003), 633-650.
doi: 10.3934/dcds.2003.9.633. |
[8] |
L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, AMS, Providence, RI, 1998. |
[9] |
F. Flori and P. Orenga, Fluid-structure interaction: Analysis of a 3-D compressible model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 753-777.
doi: 10.1016/S0294-1449(00)00119-0. |
[10] |
D. G. Gorman, I. Trendafilova, A. J. Mulholland and J. Horacek, Analytical modeling and extraction of the modal behavior of a cantilever beam in fluid interaction, Journal of Sound and Vibration, 308 (2007), 231-245. |
[11] |
C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM Journal on Mathematical Analysis, 40 (2008), 716-737.
doi: 10.1137/070699196. |
[12] |
M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid, Journal of Mathematical Fluid Mechanics, 10 (2008), 388-401.
doi: 10.1007/s00021-006-0236-4. |
[13] |
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.
doi: 10.1016/j.anihpc.2008.02.004. |
[14] |
M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure interaction model, Math. Meth. Appl. Sci., 32 (2009), 1452-1466.
doi: 10.1002/mma.1104. |
[15] |
L. Hoang and A. Ibragimov, Structural stability of generalized Forchheimer equations for compressible fluids in porous media, Nonlinearity, 24 (2011), 1-41.
doi: 10.1088/0951-7715/24/1/001. |
[16] |
J. Hron and S. Turek, A monolithic FEM/multigrid solver for ALE formulation of fluid-structure interaction with application in biomechanics, in "Fluid-structure interaction," Lecture Notes in Computational Science and Engineering, 53, Springer, Berlin, (2006), 146-170.
doi: 10.1007/3-540-34596-5_7. |
[17] |
E. Kaya, E. Aulisa, A. Ibragimov and P. Seshaiyer, A stability estimate for fluid structure interaction problem with non-linear beam, Discrete And Continuous Dynamical Systems, Supplement (2009), 424-432. |
[18] |
E. Kaya-Cekin, E. Aulisa, A. Ibragimov and P. Seshaiyer, Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam, Communications on Pure and Applied Analysis, 10 (2011), 1447-1462.
doi: 10.3934/cpaa.2011.10.1447. |
[19] |
E. Kaya-Cekin, E. Aulisa, A. Ibragimov and P. Seshaiyer, Fluid structure interaction problem with changing thickness non-linear beam, Discrete And Continuous Dynamical Systems, Supplement (2011), 813-823. |
[20] |
A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, Journal of Mathematical Analysis and Applications, 318 (2006), 92-101.
doi: 10.1016/j.jmaa.2005.05.031. |
[21] |
H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, in "Evolution Equations, Semigroups and Functional Analysis" (Milano, 2000), Progress in Nonlinear Differential Equations and their Applications, 50, Birkhäuser, Basel, (2002), 197-216. |
[22] |
I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem, Discrete And Continuous Dynamical Systems, 32 (2012), 1355-1389.
doi: 10.3934/dcds.2012.32.1355. |
[23] |
J. E. Lagnese, Modelling and stabilization of nonlinear plates, in "Estimation and Control of Distributed Parameter Systems" (Vorau, 1990), International Series of Numerical Mathematics, 100, Birkhäuser, Basel, (1991), 247-264. |
[24] |
J. E. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Studies in Applied Mathematics, 10, SIAM, Philadelphia, PA, 1989.
doi: 10.1137/1.9781611970821. |
[25] |
I. Lasiecka, Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998), 1376-1422.
doi: 10.1137/S0363012996301907. |
[26] |
V. G. Maz'ya, "Sobolev Spaces," $2^{nd}$ augmented edition, Grundlehren der mathematischen Wissenschaften, Vol. 342, Springer, 2011. |
[27] |
M. Muskat, "The Flow of Homogeneous Fluids Through Porous Media," McGraw-Hill, New York, 1937. |
[28] |
V. V. Novozhilov, "Foundations of the Nonlinear Theory of Elasticity," Dover Publication, INC, 1999. |
[29] |
J. Y. Park and J. R. Kang, Global existence and stability for a von Karman equations with memory in noncylindrical domains, Journal of Mathematical Physics, 50 (2009), 112701, 13 pp.
doi: 10.1063/1.3253977. |
[30] |
J.-P. Puel and M. Tucsnak, Boundary stabilization for the von Kármán equations, SIAM J. Control Optim., 33 (1995), 255-273.
doi: 10.1137/S0363012992228350. |
[31] |
N. Peake and S. V. Sorokin, A nonlinear model of the dynamics of a large elastic plate with heavy fluid loading, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2205-2224.
doi: 10.1098/rspa.2006.1673. |
[32] |
J. Peradze, A numerical algorithm for Kirchhoff-type nonlinear static beam, Journal of Applied Mathematics, 2009 (2009), Art. ID 818269, 12 pp.
doi: 10.1155/2009/818269. |
[33] |
M. Sathyamoorthy, "Nonlinear Analysis of Structures," CRC, 1998. |
[34] |
D. Tataru and M. Tucsnak, On the Cauchy problem for the full von Kármán system, Nonlinear Differential Equations Appl., 4 (1997), 325-340.
doi: 10.1007/s000300050018. |
[35] |
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. |
[36] |
L. Yang and C. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Analysis: Theory, Methods & Applications, 69 (2008), 3802-3810.
doi: 10.1016/j.na.2007.10.016. |
[37] |
SIAM PDE Conference 2011 San-Diego, Book of Abstracts,, Available from: , ().
|
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