# American Institute of Mathematical Sciences

October  2016, 36(10): 5445-5475. doi: 10.3934/dcds.2016040

## On one dimensional quantum Zakharov system

 1 Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan 2 Department of Mathematical Sciences, Xi'an Jiaotong-Liverpool University, SIP. Suzhou, Jiangsu, 215123, China, China

Received  April 2015 Revised  April 2016 Published  July 2016

In this paper, we discuss the properties of one dimensional quantum Zakharov system which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The system (1a)-(1b) with initial data $(E(0),n(0),\partial_t n(0))\in H^k\bigoplus H^l\bigoplus H^{l-2}$ is local well-posedness in low regularity spaces (see Theorem 1.1 and Figure 1). Especially, the low regularity result for $k$ satisfies $-3/4 < k \leq -1/4$ is obtained by using the key observation that the convoluted phase function is convex and careful bilinear analysis. The result can not be obtained by using only Strichartz inequalities for Schrödinger" waves.
Citation: Jin-Cheng Jiang, Chi-Kun Lin, Shuanglin Shao. On one dimensional quantum Zakharov system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5445-5475. doi: 10.3934/dcds.2016040
##### References:
 [1] H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equations: Smoothness and approximation,, Journal of Functional Analysis, 79 (1988), 183. doi: 10.1016/0022-1236(88)90036-5. Google Scholar [2] I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system,, J. Funct. Anal., 261 (2011), 478. doi: 10.1016/j.jfa.2011.03.015. Google Scholar [3] I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$-Schrödinger data,, Nonlinearity, 22 (2009), 1063. doi: 10.1088/0951-7715/22/5/007. Google Scholar [4] M. Ben-Artzi, H. Koch and J. C. Saut, Disperion estimates for fourth order Schrödinger equations,, C. R. Acad. Sci. Paris, 330 (2000), 87. doi: 10.1016/S0764-4442(00)00120-8. Google Scholar [5] J. Bourgain, Fourier transform restriction phenomena for certain lattic subsets and application to nonlinear evolution equations I,II,, Geom. Funct. Anal., 3 (1993), 107. doi: 10.1007/BF01895688. Google Scholar [6] J. Bourgain and J. Colliander, On wellposedness of the Zakharov system,, Internat. Math. Res. Notices, 11 (1996), 515. doi: 10.1155/S1073792896000359. Google Scholar [7] J. Colliander, J. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems,, Transactions AMS, 360 (2008), 4619. doi: 10.1090/S0002-9947-08-04295-5. Google Scholar [8] J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system,, J. Funct. Anal., 151 (1997), 384. doi: 10.1006/jfan.1997.3148. Google Scholar [9] Y. Guo, J. Zhang and B. Guo, Global well-posedness and the classical limit of the solution for the quantum Zakharov system,, Z. Angew. Math. Phys., 64 (2013), 53. doi: 10.1007/s00033-012-0215-y. Google Scholar [10] F. Haas, Quantum Plasma: An Hydrodynamic Approach,, Springer-Verlag, (2011). doi: 10.1007/978-1-4419-8201-8. Google Scholar [11] F. Haas and P. K. Shukla, Quantum and classical dynamics of Langmuir wave packets,, Physical Review E., 79 (2009), 066402. doi: 10.1103/PhysRevE.79.066402. Google Scholar [12] J.-C. Jiang, Bilinear Strichartz estimates for Schrödinger operators in 2 dimensional compact manifolds and cubic NLS,, Differential and Integral Equations, 24 (2011), 83. Google Scholar [13] C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations,, Indiana Univ. Math. J., 40 (1991), 33. doi: 10.1512/iumj.1991.40.40003. Google Scholar [14] C. Kenig, G. Ponce and L. Vega, A bilinear estimate with application to the KdV equation,, J. Amer. Math. Soc., 9 (1996), 573. doi: 10.1090/S0894-0347-96-00200-7. Google Scholar [15] N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems,, Inventiones Mathematicae, 172 (2008), 535. doi: 10.1007/s00222-008-0110-5. Google Scholar [16] T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations,, Publ. RIMS, 28 (1992), 329. doi: 10.2977/prims/1195168430. Google Scholar [17] T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations,, Differential and Integral Equations, 5 (1992), 721. Google Scholar [18] B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case,, Dynamics of PDE, 4 (2007), 197. doi: 10.4310/DPDE.2007.v4.n3.a1. Google Scholar [19] H. Schochet and M. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence,, \emph{Comm. Math. Phys.}, 106 (1986), 569. doi: 10.1007/BF01463396. Google Scholar [20] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse,, Appl. Math. Sci., 139 (1999). Google Scholar [21] V. Zakharov, Collapse of Langmuir waves,, Sov. Phys. JETP, 35 (1972), 908. Google Scholar

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##### References:
 [1] H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equations: Smoothness and approximation,, Journal of Functional Analysis, 79 (1988), 183. doi: 10.1016/0022-1236(88)90036-5. Google Scholar [2] I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system,, J. Funct. Anal., 261 (2011), 478. doi: 10.1016/j.jfa.2011.03.015. Google Scholar [3] I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$-Schrödinger data,, Nonlinearity, 22 (2009), 1063. doi: 10.1088/0951-7715/22/5/007. Google Scholar [4] M. Ben-Artzi, H. Koch and J. C. Saut, Disperion estimates for fourth order Schrödinger equations,, C. R. Acad. Sci. Paris, 330 (2000), 87. doi: 10.1016/S0764-4442(00)00120-8. Google Scholar [5] J. Bourgain, Fourier transform restriction phenomena for certain lattic subsets and application to nonlinear evolution equations I,II,, Geom. Funct. Anal., 3 (1993), 107. doi: 10.1007/BF01895688. Google Scholar [6] J. Bourgain and J. Colliander, On wellposedness of the Zakharov system,, Internat. Math. Res. Notices, 11 (1996), 515. doi: 10.1155/S1073792896000359. Google Scholar [7] J. Colliander, J. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems,, Transactions AMS, 360 (2008), 4619. doi: 10.1090/S0002-9947-08-04295-5. Google Scholar [8] J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system,, J. Funct. Anal., 151 (1997), 384. doi: 10.1006/jfan.1997.3148. Google Scholar [9] Y. Guo, J. Zhang and B. Guo, Global well-posedness and the classical limit of the solution for the quantum Zakharov system,, Z. Angew. Math. Phys., 64 (2013), 53. doi: 10.1007/s00033-012-0215-y. Google Scholar [10] F. Haas, Quantum Plasma: An Hydrodynamic Approach,, Springer-Verlag, (2011). doi: 10.1007/978-1-4419-8201-8. Google Scholar [11] F. Haas and P. K. Shukla, Quantum and classical dynamics of Langmuir wave packets,, Physical Review E., 79 (2009), 066402. doi: 10.1103/PhysRevE.79.066402. Google Scholar [12] J.-C. Jiang, Bilinear Strichartz estimates for Schrödinger operators in 2 dimensional compact manifolds and cubic NLS,, Differential and Integral Equations, 24 (2011), 83. Google Scholar [13] C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations,, Indiana Univ. Math. J., 40 (1991), 33. doi: 10.1512/iumj.1991.40.40003. Google Scholar [14] C. Kenig, G. Ponce and L. Vega, A bilinear estimate with application to the KdV equation,, J. Amer. Math. Soc., 9 (1996), 573. doi: 10.1090/S0894-0347-96-00200-7. Google Scholar [15] N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems,, Inventiones Mathematicae, 172 (2008), 535. doi: 10.1007/s00222-008-0110-5. Google Scholar [16] T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations,, Publ. RIMS, 28 (1992), 329. doi: 10.2977/prims/1195168430. Google Scholar [17] T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations,, Differential and Integral Equations, 5 (1992), 721. Google Scholar [18] B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case,, Dynamics of PDE, 4 (2007), 197. doi: 10.4310/DPDE.2007.v4.n3.a1. Google Scholar [19] H. Schochet and M. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence,, \emph{Comm. Math. Phys.}, 106 (1986), 569. doi: 10.1007/BF01463396. Google Scholar [20] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse,, Appl. Math. Sci., 139 (1999). Google Scholar [21] V. Zakharov, Collapse of Langmuir waves,, Sov. Phys. JETP, 35 (1972), 908. Google Scholar
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