The Initial Boundary Value Problem for Modified Zakharov System
ABSTRACT
In this paper, we consider the initial boudary value problem for modified Zakharov system in 3 dimensions with small initial condition. By using the continuity lemma and the linear interpolation theory, together with the properties of Sobolev spaces and the Galerkin method, we obtain the existence and uniqueness of the global solution.
Cite this paper
Li, L. and Fang, S. (2015) The Initial Boundary Value Problem for Modified Zakharov System. Advances in Pure Mathematics, 5, 278-285. doi: 10.4236/apm.2015.55028.
Li, L. and Fang, S. (2015) The Initial Boundary Value Problem for Modified Zakharov System. Advances in Pure Mathematics, 5, 278-285. doi: 10.4236/apm.2015.55028.
References
[1] Zakharov, V.E. (1972) The Collapse of Langmuir Waves. Soviet Physics—JETP, 35, 908-914. [2] Bourgain, J. and Colliander, J. (1996) On Well-Posedness of Zakharov System. International Mathematics Research Notices, 11, 515-546.
http://dx.doi.org/10.1155/S1073792896000359 [3] Ginibre, J., Tsutsumi, Y. and Velo, G. (1997) On the Cauchy Problem for Zakharov System. Journal of Functional Analysis, 151, 384-436.
http://dx.doi.org/10.1006/jfan.1997.3148 [4] Bejenaru, I., Herr, S., Holmer, J. and Tataru, D. (2009) On the 2d Zakharov System with L2 Schrodinger Data. Nonlinearity, 22, 1063-1089.
http://dx.doi.org/10.1088/0951-7715/22/5/007 [5] Colliander, J., Holmer, J. and Tzirakis, N. (2008) Low Regularity Global Well-Posedness for the Zakharov and Klein-Gordon-Schrodinger Systems. Transactions of the American Mathematical Society, 9, 4619-4638.
http://dx.doi.org/10.1090/S0002-9947-08-04295-5 [6] Pecher, H. (2005) Global Solutions with Infinite Energy for the One-Dimensional Zakharov System. Electronic Journal of Differential Equations, 41, 1-18. [7] Guo, C.H., Fang, S.M. and Guo, B.L. (2013) Long Time Behavior of the Solutions for the Dissipative Modified Zakharov Equations for Plasmas with a Quantum Correction. Journal of Mathematical Analysis and Applications, 403, 183-192.
http://dx.doi.org/10.1016/j.jmaa.2013.01.058 [8] Garcia, L.G., Haas, F., de Oliveira, L.P.L. and Goedert, J. (2005) Modified Zakaharov Equations for Plasmas with a Quantum Correction. Physics of Plasmas, 12.
http://dx.doi.org/10.1063/1.1819935 [9] You, S.J., Guo, B.L. and Ning, X.Q. (2012) Initial Boundary Value Problem for Modified Zakharov Equations. Acta Mathematica Scientia, 32B, 1455-1466. [10] You, S.J., Guo, B.L. and Ning, X.Q. (2012) Initial Boundary Value Problem for Generalized Zakharov Equations. Applied Mathematics, 57, 581-599.
http://dx.doi.org/10.1007/s10492-012-0035-6 [11] Guo, B.L., Gan, Z.H. and Zhang, J.J. (2011) Zakharov Equations and Its Solitary Wave Solutions. Science Press.
[1] Zakharov, V.E. (1972) The Collapse of Langmuir Waves. Soviet Physics—JETP, 35, 908-914. [2] Bourgain, J. and Colliander, J. (1996) On Well-Posedness of Zakharov System. International Mathematics Research Notices, 11, 515-546.
http://dx.doi.org/10.1155/S1073792896000359 [3] Ginibre, J., Tsutsumi, Y. and Velo, G. (1997) On the Cauchy Problem for Zakharov System. Journal of Functional Analysis, 151, 384-436.
http://dx.doi.org/10.1006/jfan.1997.3148 [4] Bejenaru, I., Herr, S., Holmer, J. and Tataru, D. (2009) On the 2d Zakharov System with L2 Schrodinger Data. Nonlinearity, 22, 1063-1089.
http://dx.doi.org/10.1088/0951-7715/22/5/007 [5] Colliander, J., Holmer, J. and Tzirakis, N. (2008) Low Regularity Global Well-Posedness for the Zakharov and Klein-Gordon-Schrodinger Systems. Transactions of the American Mathematical Society, 9, 4619-4638.
http://dx.doi.org/10.1090/S0002-9947-08-04295-5 [6] Pecher, H. (2005) Global Solutions with Infinite Energy for the One-Dimensional Zakharov System. Electronic Journal of Differential Equations, 41, 1-18. [7] Guo, C.H., Fang, S.M. and Guo, B.L. (2013) Long Time Behavior of the Solutions for the Dissipative Modified Zakharov Equations for Plasmas with a Quantum Correction. Journal of Mathematical Analysis and Applications, 403, 183-192.
http://dx.doi.org/10.1016/j.jmaa.2013.01.058 [8] Garcia, L.G., Haas, F., de Oliveira, L.P.L. and Goedert, J. (2005) Modified Zakaharov Equations for Plasmas with a Quantum Correction. Physics of Plasmas, 12.
http://dx.doi.org/10.1063/1.1819935 [9] You, S.J., Guo, B.L. and Ning, X.Q. (2012) Initial Boundary Value Problem for Modified Zakharov Equations. Acta Mathematica Scientia, 32B, 1455-1466. [10] You, S.J., Guo, B.L. and Ning, X.Q. (2012) Initial Boundary Value Problem for Generalized Zakharov Equations. Applied Mathematics, 57, 581-599.
http://dx.doi.org/10.1007/s10492-012-0035-6 [11] Guo, B.L., Gan, Z.H. and Zhang, J.J. (2011) Zakharov Equations and Its Solitary Wave Solutions. Science Press.