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 and Continuous Dynamical Systems, 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), 183-210. doi: 10.1016/0022-1236(88)90036-5.

[2]

I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system, J. Funct. Anal., 261 (2011), 478-506. doi: 10.1016/j.jfa.2011.03.015.

[3]

I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$-Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. doi: 10.1088/0951-7715/22/5/007.

[4]

M. Ben-Artzi, H. Koch and J. C. Saut, Disperion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris, Série I, 330 (2000), 87-92. doi: 10.1016/S0764-4442(00)00120-8.

[5]

J. Bourgain, Fourier transform restriction phenomena for certain lattic subsets and application to nonlinear evolution equations I,II, Geom. Funct. Anal., 3 (1993), 107-156, 209-262. doi: 10.1007/BF01895688.

[6]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. Math. Res. Notices, 11 (1996), 515-546. doi: 10.1155/S1073792896000359.

[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), 4619-4638. doi: 10.1090/S0002-9947-08-04295-5.

[8]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.

[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), 53-68. doi: 10.1007/s00033-012-0215-y.

[10]

F. Haas, Quantum Plasma: An Hydrodynamic Approach, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-8201-8.

[11]

F. Haas and P. K. Shukla, Quantum and classical dynamics of Langmuir wave packets, Physical Review E., 79 (2009), 066402-066501. doi: 10.1103/PhysRevE.79.066402.

[12]

J.-C. Jiang, Bilinear Strichartz estimates for Schrödinger operators in 2 dimensional compact manifolds and cubic NLS, Differential and Integral Equations, 24 (2011), 83-108.

[13]

C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003.

[14]

C. Kenig, G. Ponce and L. Vega, A bilinear estimate with application to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.

[15]

N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems, Inventiones Mathematicae, 172 (2008), 535-583. doi: 10.1007/s00222-008-0110-5.

[16]

T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. RIMS, Kyoto Univ., 28 (1992), 329-361. doi: 10.2977/prims/1195168430.

[17]

T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Differential and Integral Equations, 5 (1992), 721-745.

[18]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225. doi: 10.4310/DPDE.2007.v4.n3.a1.

[19]

H. Schochet and M. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys., 106 (1986), 569-580. doi: 10.1007/BF01463396.

[20]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse, Appl. Math. Sci., Vol.139, Springer-Verlag, New York, 1999.

[21]

V. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.

show all references

References:
[1]

H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equations: Smoothness and approximation, Journal of Functional Analysis, 79 (1988), 183-210. doi: 10.1016/0022-1236(88)90036-5.

[2]

I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system, J. Funct. Anal., 261 (2011), 478-506. doi: 10.1016/j.jfa.2011.03.015.

[3]

I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$-Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. doi: 10.1088/0951-7715/22/5/007.

[4]

M. Ben-Artzi, H. Koch and J. C. Saut, Disperion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris, Série I, 330 (2000), 87-92. doi: 10.1016/S0764-4442(00)00120-8.

[5]

J. Bourgain, Fourier transform restriction phenomena for certain lattic subsets and application to nonlinear evolution equations I,II, Geom. Funct. Anal., 3 (1993), 107-156, 209-262. doi: 10.1007/BF01895688.

[6]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. Math. Res. Notices, 11 (1996), 515-546. doi: 10.1155/S1073792896000359.

[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), 4619-4638. doi: 10.1090/S0002-9947-08-04295-5.

[8]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.

[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), 53-68. doi: 10.1007/s00033-012-0215-y.

[10]

F. Haas, Quantum Plasma: An Hydrodynamic Approach, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-8201-8.

[11]

F. Haas and P. K. Shukla, Quantum and classical dynamics of Langmuir wave packets, Physical Review E., 79 (2009), 066402-066501. doi: 10.1103/PhysRevE.79.066402.

[12]

J.-C. Jiang, Bilinear Strichartz estimates for Schrödinger operators in 2 dimensional compact manifolds and cubic NLS, Differential and Integral Equations, 24 (2011), 83-108.

[13]

C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003.

[14]

C. Kenig, G. Ponce and L. Vega, A bilinear estimate with application to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.

[15]

N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems, Inventiones Mathematicae, 172 (2008), 535-583. doi: 10.1007/s00222-008-0110-5.

[16]

T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. RIMS, Kyoto Univ., 28 (1992), 329-361. doi: 10.2977/prims/1195168430.

[17]

T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Differential and Integral Equations, 5 (1992), 721-745.

[18]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225. doi: 10.4310/DPDE.2007.v4.n3.a1.

[19]

H. Schochet and M. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys., 106 (1986), 569-580. doi: 10.1007/BF01463396.

[20]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse, Appl. Math. Sci., Vol.139, Springer-Verlag, New York, 1999.

[21]

V. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.

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