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Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations
1. | Kharkov National Universit, Department of Mathematics and Mechanics, 4 Svobody sq, 61077 Kharkov |
2. | Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody Sq. 61077 Kharkov, Ukraine |
References:
[1] |
M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger systems, Differential Integral Equations, 16 (2003), 573-581. |
[2] |
A. Babin and M. Vishik, "Attractors of Evolution Equations," Translated and revised from the 1989 Russian original by Babin, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[3] |
J. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Continuous Dynam. Systems, 10 (2004), 31-52. |
[4] |
P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212.
doi: 10.1137/0521065. |
[5] |
J. Bourgain, On the Cauchy and invariant measure proplem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202.
doi: 10.1215/S0012-7094-94-07607-2. |
[6] |
J. Bourgain and J. Colliander, On well-posedness of the Zakharov system,, Internat. Math. Res. Notices, 1996 (): 515.
doi: 10.1155/S1073792896000359. |
[7] |
H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681.
doi: 10.1016/0362-546X(80)90068-1. |
[8] |
A. Boutet de Monvel and I. Chueshov, Uniqueness theorem for weak solutions of von Karman evolution equations, J. Mathematical Analysis and Applications, 221 (1998), 419-429.
doi: 10.1006/jmaa.1997.5681. |
[9] |
I. D. Chueshov, "Vvedenie v Teoriyu Beskonechnomernykh Dissipativnykh Sistem," (Russian) [Introduction to the Theory of Infinite-Dimensional Dissipative Systems], Universitet·skie Lektsii po Sovremennoĭ Matematike [University Lectures in Contemporary Mathematics], AKTA, Kharkiv, 1999. English translation in Acta, Kharkov, 2002; see also http://www.emis.de/monographs/Chueshov/. |
[10] |
I. Chueshov and I. Lasiecka, Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Discrete Continuous Dynam. Systems, 15 (2006), 777-809. |
[11] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. |
[12] |
I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics," Springer Monographs in Mathematics, Springer, New York, 2010. |
[13] |
I. Chueshov and I. Lasiecka, On Global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Commun. Partial Dif. Eqs., 36 (2011), 67-99.
doi: 10.1080/03605302.2010.484472. |
[14] |
I. Chueshov and A. Shcherbina, On 2D Zakharov system in a bounded domain, Differential and Integral Equations, 18 (2005), 781-812. |
[15] |
I. Flahaut, Attractors for the dissipative Zakharov system, Nonlinear Analysis, 16 (1991), 599-633.
doi: 10.1016/0362-546X(91)90170-6. |
[16] |
J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy proplem for the Zakharov system, J. Functional Analysis, 151 (1997), 384-436.
doi: 10.1006/jfan.1997.3148. |
[17] |
L. Glangetas and F. Merle, Existence and self-similar blow up solutions for Zakharov equation in dimension two. I, Commun. Math. Phys., 160 (1994), 173-215.
doi: 10.1007/BF02099792. |
[18] |
O. Goubet and I. Moise, Attractor for dissipative Zakharov system, Nonlinear Analysis, 31 (1998), 823-847.
doi: 10.1016/S0362-546X(97)00441-0. |
[19] |
M. Grasselli, G. Schimperna, and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term, Commun. Partial Dif. Eqs., 34 (2009), 137-170.
doi: 10.1080/03605300802608247. |
[20] |
B. Guo and F. Chen, Finite dimensional behaviour of global attractors for weakly damped nonlinear Schrödinger-Boussinesq equations, Physica D, 93 (1996), 101-118 |
[21] |
B. Guo and X. Du, Existence of the time periodic solution for damped Schrödinger-Boussinesq equation, Commun. in Nonlin. Sci. Numer. Simul., 5 (2000), 179-183.
doi: 10.1016/S1007-5704(00)90032-7. |
[22] |
B. Guo and X. Du, The behavior of attractors for damped Schrödinger-Boussinesq equation, Commun. in Nonlin. Sci. Numer. Simul., 6 (2001), 54-60.
doi: 10.1016/S1007-5704(01)90030-9. |
[23] |
B. Guo and X. Du, Existence of the periodic solution for the weakly damped Schrödinger-Boussinesq equation, J. Mathematical Analysis and Applications, 262 (2001), 453-472.
doi: 10.1006/jmaa.2000.7455. |
[24] |
N. Hayashi and W. von Wahl, On the global strong solutions of coupled Klein-Gordon-Schrödinger equations, J. Mathematical Society of Japan, 39 (1987), 489-497.
doi: 10.2969/jmsj/03930489. |
[25] |
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. |
[26] |
O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Lezioni Lincee [Lincei Lectures], Cambridge University Press, Cambridge, 1991. |
[27] |
Y. Li and Q. Chen, Finite dimensional global attractor for dissipative Schrödinger-Boussinesq equations, J. Mathematical Analysis and Applications, 205 (1997), 107-132.
doi: 10.1006/jmaa.1996.5148. |
[28] |
J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications," Vol. 2, (French), Travaux et Recherches Mathématiques, No. 18, Dunod, Paris, 1968. |
[29] |
K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Differential Equations, 170 (2001), 281-316. |
[30] |
I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393.
doi: 10.1088/0951-7715/11/5/012. |
[31] |
V. Sedenko, Uniqueness of generalized solution of initial boundary value problem of nonlinear oscilations theory of shallow shells, (Russian), Dokl. Akad. Nauk SSSR, 316 (1991), 1319-1322; translation in Soviet Math. Dokl., 43 (1991), 284–-287. |
[32] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata (4), 146 (1987), 65-96. |
[33] |
A. Shcherbina, Gevrey regularity of the global attractor for the dissipative Zakharov system, Dynamical Systems, 18 (2003), 201-225.
doi: 10.1080/14689360310001597269. |
[34] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. |
show all references
References:
[1] |
M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger systems, Differential Integral Equations, 16 (2003), 573-581. |
[2] |
A. Babin and M. Vishik, "Attractors of Evolution Equations," Translated and revised from the 1989 Russian original by Babin, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[3] |
J. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Continuous Dynam. Systems, 10 (2004), 31-52. |
[4] |
P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212.
doi: 10.1137/0521065. |
[5] |
J. Bourgain, On the Cauchy and invariant measure proplem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202.
doi: 10.1215/S0012-7094-94-07607-2. |
[6] |
J. Bourgain and J. Colliander, On well-posedness of the Zakharov system,, Internat. Math. Res. Notices, 1996 (): 515.
doi: 10.1155/S1073792896000359. |
[7] |
H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681.
doi: 10.1016/0362-546X(80)90068-1. |
[8] |
A. Boutet de Monvel and I. Chueshov, Uniqueness theorem for weak solutions of von Karman evolution equations, J. Mathematical Analysis and Applications, 221 (1998), 419-429.
doi: 10.1006/jmaa.1997.5681. |
[9] |
I. D. Chueshov, "Vvedenie v Teoriyu Beskonechnomernykh Dissipativnykh Sistem," (Russian) [Introduction to the Theory of Infinite-Dimensional Dissipative Systems], Universitet·skie Lektsii po Sovremennoĭ Matematike [University Lectures in Contemporary Mathematics], AKTA, Kharkiv, 1999. English translation in Acta, Kharkov, 2002; see also http://www.emis.de/monographs/Chueshov/. |
[10] |
I. Chueshov and I. Lasiecka, Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Discrete Continuous Dynam. Systems, 15 (2006), 777-809. |
[11] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. |
[12] |
I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics," Springer Monographs in Mathematics, Springer, New York, 2010. |
[13] |
I. Chueshov and I. Lasiecka, On Global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Commun. Partial Dif. Eqs., 36 (2011), 67-99.
doi: 10.1080/03605302.2010.484472. |
[14] |
I. Chueshov and A. Shcherbina, On 2D Zakharov system in a bounded domain, Differential and Integral Equations, 18 (2005), 781-812. |
[15] |
I. Flahaut, Attractors for the dissipative Zakharov system, Nonlinear Analysis, 16 (1991), 599-633.
doi: 10.1016/0362-546X(91)90170-6. |
[16] |
J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy proplem for the Zakharov system, J. Functional Analysis, 151 (1997), 384-436.
doi: 10.1006/jfan.1997.3148. |
[17] |
L. Glangetas and F. Merle, Existence and self-similar blow up solutions for Zakharov equation in dimension two. I, Commun. Math. Phys., 160 (1994), 173-215.
doi: 10.1007/BF02099792. |
[18] |
O. Goubet and I. Moise, Attractor for dissipative Zakharov system, Nonlinear Analysis, 31 (1998), 823-847.
doi: 10.1016/S0362-546X(97)00441-0. |
[19] |
M. Grasselli, G. Schimperna, and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term, Commun. Partial Dif. Eqs., 34 (2009), 137-170.
doi: 10.1080/03605300802608247. |
[20] |
B. Guo and F. Chen, Finite dimensional behaviour of global attractors for weakly damped nonlinear Schrödinger-Boussinesq equations, Physica D, 93 (1996), 101-118 |
[21] |
B. Guo and X. Du, Existence of the time periodic solution for damped Schrödinger-Boussinesq equation, Commun. in Nonlin. Sci. Numer. Simul., 5 (2000), 179-183.
doi: 10.1016/S1007-5704(00)90032-7. |
[22] |
B. Guo and X. Du, The behavior of attractors for damped Schrödinger-Boussinesq equation, Commun. in Nonlin. Sci. Numer. Simul., 6 (2001), 54-60.
doi: 10.1016/S1007-5704(01)90030-9. |
[23] |
B. Guo and X. Du, Existence of the periodic solution for the weakly damped Schrödinger-Boussinesq equation, J. Mathematical Analysis and Applications, 262 (2001), 453-472.
doi: 10.1006/jmaa.2000.7455. |
[24] |
N. Hayashi and W. von Wahl, On the global strong solutions of coupled Klein-Gordon-Schrödinger equations, J. Mathematical Society of Japan, 39 (1987), 489-497.
doi: 10.2969/jmsj/03930489. |
[25] |
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. |
[26] |
O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Lezioni Lincee [Lincei Lectures], Cambridge University Press, Cambridge, 1991. |
[27] |
Y. Li and Q. Chen, Finite dimensional global attractor for dissipative Schrödinger-Boussinesq equations, J. Mathematical Analysis and Applications, 205 (1997), 107-132.
doi: 10.1006/jmaa.1996.5148. |
[28] |
J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications," Vol. 2, (French), Travaux et Recherches Mathématiques, No. 18, Dunod, Paris, 1968. |
[29] |
K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Differential Equations, 170 (2001), 281-316. |
[30] |
I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393.
doi: 10.1088/0951-7715/11/5/012. |
[31] |
V. Sedenko, Uniqueness of generalized solution of initial boundary value problem of nonlinear oscilations theory of shallow shells, (Russian), Dokl. Akad. Nauk SSSR, 316 (1991), 1319-1322; translation in Soviet Math. Dokl., 43 (1991), 284–-287. |
[32] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata (4), 146 (1987), 65-96. |
[33] |
A. Shcherbina, Gevrey regularity of the global attractor for the dissipative Zakharov system, Dynamical Systems, 18 (2003), 201-225.
doi: 10.1080/14689360310001597269. |
[34] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. |
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