Taiwanese Journal of Mathematics

Semi-classical Limit for the Quantum Zakharov System

Abstract

In this paper, we prove the semi-classical limit for the quantum Zakharov system, that is, the quantum Zakharov system converges to the classical Zakharov system as the quantum parameter goes to zero, including a convergence rate. We improve the results of Guo-Zhang-Guo [11].

Article information

Source
Taiwanese J. Math., Volume 23, Number 4 (2019), 925-949.

Dates
Revised: 7 August 2018
Accepted: 13 August 2018
First available in Project Euclid: 18 July 2019

https://projecteuclid.org/euclid.twjm/1563436875

Digital Object Identifier
doi:10.11650/tjm/180806

Mathematical Reviews number (MathSciNet)
MR3982068

Zentralblatt MATH identifier
07088954

Citation

Fang, Yung-Fu; Kuo, Hung-Wen; Shih, Hsi-Wei; Wang, Kuan-Hsiang. Semi-classical Limit for the Quantum Zakharov System. Taiwanese J. Math. 23 (2019), no. 4, 925--949. doi:10.11650/tjm/180806. https://projecteuclid.org/euclid.twjm/1563436875

References

• J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. Math. Res. Notices 1996 (1996), no. 11, 515–546.
• T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, American Mathematical Society, Providence, RI, 2003.
• T.-J. Chen, Y.-F. Fang and K.-H. Wang, Low regularity global well-posedness for the quantum Zakharov system in $1D$, Taiwanese J. Math. 21 (2017), no. 2, 341–361.
• J. Colliander, J. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Math. Soc. 360 (2008), no. 9, 4619–4638.
• Y.-F. Fang, H.-W. Shih and K.-H. Wang, Local well-posedness for the quantum Zakharov system in one spatial dimension, J. Hyperbolic Differ. Equ. 14 (2017), no. 1, 157–192.
• L. G. Garcia, F. Haas, L. P. L. de Oliveira and J. Goedert, Modified Zakharov equations for plasmas with a quantum correction, Phys. Plasmas 12 (2005), no. 1, 012302.
• J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151 (1997), no. 2, 384–436.
• L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two I, Comm. Math. Phys. 160 (1994), no. 1, 173–215.
• ––––, Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two II, Comm. Math. Phys. 160 (1994), no. 2, 349–389.
• Z. Guo, K. Nakanishi and S. Wang, Global dynamics below the ground state energy for the Zakharov system in the 3D radial case, Adv. Math. 238 (2013), 412–441.
• 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), no. 1, 53–68.
• S. Gustafson, K. Nakanishi and T.-P. Tsai, Scattering for the Gross-Pitaevskii equation, Math. Res. Lett. 13 (2006), no. 2-3, 273–285.
• F. Haas, Quantum Plasmas: An hydrodynamic approach, Springer Series on Atomics, Optical, and Plasma Physics 65, Springer, New York, 2011.
• F. Haas and P. K. Shukla, Quantum and classical dynamics of Langmuir wave packets, Phys. Rev. E 79 (2009), no. 6, 066402.
• J.-C. Jiang, C.-K. Lin and S. Shao, On one dimensional quantum Zakharov system, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5445–5475.
• C. E. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct. Anal. 127 (1995), no. 1, 204–234.
• N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems, Invent. Math. 172 (2008), no. 3, 535–583.
• T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci. 28 (1992), no. 3, 329–361.
• ––––, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Differential Integral Equations 5 (1992), no. 4, 721–745.
• B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ. 4 (2007), no. 3, 197–225.
• S. H. Schochet and M. I. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys. 106 (1986), no. 4, 569–580.