Multilinear weighted estimates and quantum Zakharov system
Abstract
We consider the well-posedness theory of the compact case of one-dimensional quantum Zakharov system with the periodic boundary condition. The global well-posedness for sufficiently regular data is shown. The semi-classical limit as is obtained on a compact time interval whereas the quantum perturbation proves to be singular on an infinite time interval.
Keyword : quantum Zakharov system, well-posedness, higher order perturbation
How to Cite
Choi, B. (2022). Multilinear weighted estimates and quantum Zakharov system. Mathematical Modelling and Analysis, 27(2), 342–359. https://doi.org/10.3846/mma.2022.15555
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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V.E. Zakharov et al. Collapse of Langmuir waves. Sov. Phys. JETP, 35(5):908–914, 1972.
J. Bourgain. On the Cauchy and invariant measure problem for the periodic Zakharov system. Duke Math. J, 76(1):175–202, 1994. https://doi.org/10.1215/S0012-7094-94-07607-2
T.-J. Chen, Y.-F. Fang and K.-H. Wang. Low regularity global well-posedness for the quantum Zakharov system in 1D. Taiwanese Journal of Mathematics, 21(2):341–361, 2017. https://doi.org/10.11650/tjm/7806
M.B. Erdog˘an and N. Tzirakis. Smoothing and global attractors for the Zakharov system on the torus. Analysis & PDE, 6(3):723–750, 2013. https://doi.org/10.2140/apde.2013.6.723
Y.-F. Fang, H.-W. Shih and K.-H. Wang. Local well-posedness for the quantum Zakharov system in one spatial dimension. Journal of Hyperbolic Differential Equations, 14(01):157–192, 2017. https://doi.org/10.1142/S0219891617500059
L.G. Garcia, F. Haas, L.P.L. De Oliveira and J. Goedert. Modified Zakharov equations for plasmas with a quantum correction. Physics of Plasmas, 12(1):012302, 2005. https://doi.org/10.1063/1.1819935
J. Ginibre, Y. Tsutsumi and G. Velo. On the Cauchy problem for the Zakharov system. Journal of Functional Analysis, 151(2):384–436, 1997. https://doi.org/10.1006/jfan.1997.3148
Y. Guo, J. Zhang and B. Guo. Global well-posedness and the classical limit of the solution for the quantum Zakharov system. Zeitschrift fu¨r angewandte Mathematik und Physik, 64(1):53–68, 2013. https://doi.org/10.1007/s00033-0120215-y
F. Haas. Quantum plasmas: An hydrodynamic approach, volume 65. Springer Science & Business Media, 2011.
J.-C. Jiang, C.-K. Lin and S. Shao. On one dimensional quantum Zakharov system. Discrete & Continuous Dynamical Systems, 36(10):5445–5475, 2016. https://doi.org/10.3934/dcds.2016040
C. Kenig, G. Ponce and L. Vega. A bilinear estimate with applications to the KdV equation. Journal of the American Mathematical Society, 9(2):573–603, 1996. https://doi.org/10.1090/S0894-0347-96-00200-7
C. Kenig, G. Ponce and L. Vega. Quadratic forms for the 1-D semilinear Schr¨odinger equation. Transactions of the American Mathematical Society, 348(8):3323–3353, 1996. https://doi.org/10.1090/S0002-9947-96-01645-5
N. Kishimoto. Local well-posedness for the Zakharov system on the multidimensional torus. Journal d’Analyse Math´ematique, 119(1):213–253, 2013. https://doi.org/10.1007/s11854-013-0007-0
G. Simpson, C. Sulem and P.-L. Sulem. Arrest of Langmuir wave collapse by quantum effects. Physical Review E, 80(5):056405, 2009. https://doi.org/10.1103/PhysRevE.80.056405
H. Takaoka et al. Well-posedness for the Zakharov system with the periodic boundary condition. Differential and Integral Equations, 12(6):789–810, 1999.
T. Tao. Nonlinear dispersive equations: local and global analysis. Number 106. American Mathematical Soc., 2006.
V.E. Zakharov et al. Collapse of Langmuir waves. Sov. Phys. JETP, 35(5):908–914, 1972.