# American Institute of Mathematical Sciences

June  2012, 1(1): 57-80. doi: 10.3934/eect.2012.1.57

## 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

Received  October 2011 Revised  January 2012 Published  March 2012

We deal with an initial boundary value problem for the Schrödinger-Boussinesq system arising in plasma physics in two-dimensional domains. We prove the global Hadamard well-posedness of this problem (with respect to the topology which is weaker than topology associated with the standard variational (weak) solutions) and study properties of the solutions. In the dissipative case the existence of a global attractor is established.
Citation: Igor Chueshov, Alexey Shcherbina. Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations. Evolution Equations & Control Theory, 2012, 1 (1) : 57-80. doi: 10.3934/eect.2012.1.57
##### 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.   Google Scholar [2] A. Babin and M. Vishik, "Attractors of Evolution Equations,", Translated and revised from the 1989 Russian original by Babin, 25 (1989).   Google Scholar [3] J. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications,, Discrete Continuous Dynam. Systems, 10 (2004), 31.   Google Scholar [4] P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling,, SIAM J. Math. Anal., 21 (1990), 1190.  doi: 10.1137/0521065.  Google Scholar [5] J. Bourgain, On the Cauchy and invariant measure proplem for the periodic Zakharov system,, Duke Math. J., 76 (1994), 175.  doi: 10.1215/S0012-7094-94-07607-2.  Google Scholar [6] J. Bourgain and J. Colliander, On well-posedness of the Zakharov system,, Internat. Math. Res. Notices, 1996 (): 515.  doi: 10.1155/S1073792896000359.  Google Scholar [7] H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677.  doi: 10.1016/0362-546X(80)90068-1.  Google Scholar [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.  doi: 10.1006/jmaa.1997.5681.  Google Scholar [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], (1999).   Google Scholar [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.   Google Scholar [11] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008).   Google Scholar [12] I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics,", Springer Monographs in Mathematics, (2010).   Google Scholar [13] I. Chueshov and I. Lasiecka, On Global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity,, Commun. Partial Dif. Eqs., 36 (2011), 67.  doi: 10.1080/03605302.2010.484472.  Google Scholar [14] I. Chueshov and A. Shcherbina, On 2D Zakharov system in a bounded domain,, Differential and Integral Equations, 18 (2005), 781.   Google Scholar [15] I. Flahaut, Attractors for the dissipative Zakharov system,, Nonlinear Analysis, 16 (1991), 599.  doi: 10.1016/0362-546X(91)90170-6.  Google Scholar [16] J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy proplem for the Zakharov system,, J. Functional Analysis, 151 (1997), 384.  doi: 10.1006/jfan.1997.3148.  Google Scholar [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.  doi: 10.1007/BF02099792.  Google Scholar [18] O. Goubet and I. Moise, Attractor for dissipative Zakharov system,, Nonlinear Analysis, 31 (1998), 823.  doi: 10.1016/S0362-546X(97)00441-0.  Google Scholar [19] M. Grasselli, G. Schimperna, and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, Commun. Partial Dif. Eqs., 34 (2009), 137.  doi: 10.1080/03605300802608247.  Google Scholar [20] B. Guo and F. Chen, Finite dimensional behaviour of global attractors for weakly damped nonlinear Schrödinger-Boussinesq equations,, Physica D, 93 (1996), 101.   Google Scholar [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.  doi: 10.1016/S1007-5704(00)90032-7.  Google Scholar [22] B. Guo and X. Du, The behavior of attractors for damped Schrödinger-Boussinesq equation,, Commun. in Nonlin. Sci. Numer. Simul., 6 (2001), 54.  doi: 10.1016/S1007-5704(01)90030-9.  Google Scholar [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.  doi: 10.1006/jmaa.2000.7455.  Google Scholar [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.  doi: 10.2969/jmsj/03930489.  Google Scholar [25] H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems,, in, 50 (2002), 197.   Google Scholar [26] O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,", Lezioni Lincee [Lincei Lectures], (1991).   Google Scholar [27] Y. Li and Q. Chen, Finite dimensional global attractor for dissipative Schrödinger-Boussinesq equations,, J. Mathematical Analysis and Applications, 205 (1997), 107.  doi: 10.1006/jmaa.1996.5148.  Google Scholar [28] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications," Vol. 2, (French),, Travaux et Recherches Mathématiques, (1968).   Google Scholar [29] K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains,, J. Differential Equations, 170 (2001), 281.   Google Scholar [30] I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations,, Nonlinearity, 11 (1998), 1369.  doi: 10.1088/0951-7715/11/5/012.  Google Scholar [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.   Google Scholar [32] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Annali di Matematica Pura ed Applicata (4), 146 (1987), 65.   Google Scholar [33] A. Shcherbina, Gevrey regularity of the global attractor for the dissipative Zakharov system,, Dynamical Systems, 18 (2003), 201.  doi: 10.1080/14689360310001597269.  Google Scholar [34] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Applied Mathematical Sciences, 68 (1988).   Google Scholar

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.   Google Scholar [2] A. Babin and M. Vishik, "Attractors of Evolution Equations,", Translated and revised from the 1989 Russian original by Babin, 25 (1989).   Google Scholar [3] J. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications,, Discrete Continuous Dynam. Systems, 10 (2004), 31.   Google Scholar [4] P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling,, SIAM J. Math. Anal., 21 (1990), 1190.  doi: 10.1137/0521065.  Google Scholar [5] J. Bourgain, On the Cauchy and invariant measure proplem for the periodic Zakharov system,, Duke Math. J., 76 (1994), 175.  doi: 10.1215/S0012-7094-94-07607-2.  Google Scholar [6] J. Bourgain and J. Colliander, On well-posedness of the Zakharov system,, Internat. Math. Res. Notices, 1996 (): 515.  doi: 10.1155/S1073792896000359.  Google Scholar [7] H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677.  doi: 10.1016/0362-546X(80)90068-1.  Google Scholar [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.  doi: 10.1006/jmaa.1997.5681.  Google Scholar [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], (1999).   Google Scholar [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.   Google Scholar [11] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008).   Google Scholar [12] I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics,", Springer Monographs in Mathematics, (2010).   Google Scholar [13] I. Chueshov and I. Lasiecka, On Global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity,, Commun. Partial Dif. Eqs., 36 (2011), 67.  doi: 10.1080/03605302.2010.484472.  Google Scholar [14] I. Chueshov and A. Shcherbina, On 2D Zakharov system in a bounded domain,, Differential and Integral Equations, 18 (2005), 781.   Google Scholar [15] I. Flahaut, Attractors for the dissipative Zakharov system,, Nonlinear Analysis, 16 (1991), 599.  doi: 10.1016/0362-546X(91)90170-6.  Google Scholar [16] J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy proplem for the Zakharov system,, J. Functional Analysis, 151 (1997), 384.  doi: 10.1006/jfan.1997.3148.  Google Scholar [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.  doi: 10.1007/BF02099792.  Google Scholar [18] O. Goubet and I. Moise, Attractor for dissipative Zakharov system,, Nonlinear Analysis, 31 (1998), 823.  doi: 10.1016/S0362-546X(97)00441-0.  Google Scholar [19] M. Grasselli, G. Schimperna, and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, Commun. Partial Dif. Eqs., 34 (2009), 137.  doi: 10.1080/03605300802608247.  Google Scholar [20] B. Guo and F. Chen, Finite dimensional behaviour of global attractors for weakly damped nonlinear Schrödinger-Boussinesq equations,, Physica D, 93 (1996), 101.   Google Scholar [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.  doi: 10.1016/S1007-5704(00)90032-7.  Google Scholar [22] B. Guo and X. Du, The behavior of attractors for damped Schrödinger-Boussinesq equation,, Commun. in Nonlin. Sci. Numer. Simul., 6 (2001), 54.  doi: 10.1016/S1007-5704(01)90030-9.  Google Scholar [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.  doi: 10.1006/jmaa.2000.7455.  Google Scholar [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.  doi: 10.2969/jmsj/03930489.  Google Scholar [25] H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems,, in, 50 (2002), 197.   Google Scholar [26] O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,", Lezioni Lincee [Lincei Lectures], (1991).   Google Scholar [27] Y. Li and Q. Chen, Finite dimensional global attractor for dissipative Schrödinger-Boussinesq equations,, J. Mathematical Analysis and Applications, 205 (1997), 107.  doi: 10.1006/jmaa.1996.5148.  Google Scholar [28] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications," Vol. 2, (French),, Travaux et Recherches Mathématiques, (1968).   Google Scholar [29] K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains,, J. Differential Equations, 170 (2001), 281.   Google Scholar [30] I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations,, Nonlinearity, 11 (1998), 1369.  doi: 10.1088/0951-7715/11/5/012.  Google Scholar [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.   Google Scholar [32] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Annali di Matematica Pura ed Applicata (4), 146 (1987), 65.   Google Scholar [33] A. Shcherbina, Gevrey regularity of the global attractor for the dissipative Zakharov system,, Dynamical Systems, 18 (2003), 201.  doi: 10.1080/14689360310001597269.  Google Scholar [34] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Applied Mathematical Sciences, 68 (1988).   Google Scholar
 [1] Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777 [2] Chunhua Li. Decay of solutions for a system of nonlinear Schrödinger equations in 2D. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4265-4285. doi: 10.3934/dcds.2012.32.4265 [3] Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3737-3765. doi: 10.3934/dcds.2019230 [4] Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37 [5] Sevdzhan Hakkaev. Orbital stability of solitary waves of the Schrödinger-Boussinesq equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1043-1050. doi: 10.3934/cpaa.2007.6.1043 [6] Tetsu Mizumachi. Instability of bound states for 2D nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 413-428. doi: 10.3934/dcds.2005.13.413 [7] Wenru Huo, Aimin Huang. The global attractor of the 2d Boussinesq equations with fractional Laplacian in subcritical case. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2531-2550. doi: 10.3934/dcdsb.2016059 [8] Renhui Wan. Global well-posedness for the 2D Boussinesq equations with a velocity damping term. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2709-2730. doi: 10.3934/dcds.2019113 [9] Gianluca Crippa, Elizaveta Semenova, Stefano Spirito. Strong continuity for the 2D Euler equations. Kinetic & Related Models, 2015, 8 (4) : 685-689. doi: 10.3934/krm.2015.8.685 [10] Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183 [11] Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165 [12] Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $L^2$-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122 [13] Fei Hou, Huicheng Yin. On global axisymmetric solutions to 2D compressible full Euler equations of Chaplygin gases. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1435-1492. doi: 10.3934/dcds.2020083 [14] Brian Ryals, Robert J. Sacker. Global stability in the 2D Ricker equation revisited. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 585-604. doi: 10.3934/dcdsb.2017028 [15] Bo-Qing Dong, Jiahong Wu, Xiaojing Xu, Zhuan Ye. Global regularity for the 2D micropolar equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4133-4162. doi: 10.3934/dcds.2018180 [16] Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921 [17] Leonardo Kosloff, Tomas Schonbek. Existence and decay of solutions of the 2D QG equation in the presence of an obstacle. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1025-1043. doi: 10.3934/dcdss.2014.7.1025 [18] María J. Martín, Jukka Tuomela. 2D incompressible Euler equations: New explicit solutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4547-4563. doi: 10.3934/dcds.2019187 [19] Kazumasa Fujiwara, Tohru Ozawa. On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 275-280. doi: 10.3934/eect.2018013 [20] Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068

2018 Impact Factor: 1.048