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論文名稱(中文) 關於一維的廣義量子Zakharov系統的局部適定性結果
論文名稱(英文) A result of local well-posedness for the general quantum Zakharov system in one dimension
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 108
學期 2
出版年 109
研究生(中文) 王品瑜
研究生(英文) Pin-Yu Wang
學號 L16064020
學位類別 碩士
語文別 英文
論文頁數 76頁
口試委員 指導教授-史習偉
口試委員-方永富
口試委員-郭鴻文
中文關鍵字 Zakharov 系統  量子 Zakharov 系統  general Zakharov 系統  general 量子 Zakharov 系統  局部適定性  Strichartz 估計  四階薛丁格方程  四階波方程 
英文關鍵字 Zakharov system  quantum Zakharov system  general Zakharov system  general quantum Zakharov system  local well-posedness  Strichartz estimates  fourth-order Schrödinger equation  fourth-order wave equation 
學科別分類
中文摘要 在這篇論文裡我們探討了一維的 general quantum Zakharov system 在一個 distinguishied polynomial 狀況下的局部適定性問題,見(1.1)-(1.2)。此系統是接續 [9] 以 及 [6] 的延伸,我們將使用 Bourgain space,[10] 中使用的方法,[8, 15] 拓展 [10] 的一些工具,給出 QZSγ, γ = 1.5,Schrödinger 方程捨掉絕對值狀況的局部適定性範圍。
英文摘要 In this paper, we consider the local well-posedness problem for the general quantum Zakharov system in the distinguished polynomial case in 1-dimension. This system is extended from [9] and [6]. Use Bourgain space, the method in [10], and some tools in [8, 15] which was extended form [10], we will give the well-posed region for QZSγ in γ = 1.5. Schrödinger part considered here do not with an absolute value like what appears in Schrödinger equation of QZSγ.
論文目次 1 Introduction . . . 1

2 Notation . . . 6

3 Local Well-Posedness for General Quantum Zakharov System . . . 8
3.1 Solution formulas, linear estimates, and some other tools . . . 8
3.2 Multilinear Estimates . . . 13
3.3 Multilinear Estimates for Schrödinger Part . . . 17
3.4 Multilinear Estimates for Wave Part . . . 53
3.5 MainResults . . . 70

References . . . 75
參考文獻 [1] H. Added and S. Added, Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris 299, 551-554, 1984.

[2] Bourgain, Jean, and J. Colliander. ”On wellposedness of the Zakharov system.” Inter- national Mathematics Research Notices 1996.11, 515-546, 1996.

[3] Bejenaru, I., Guo, Z., Herr, S., and Nakanishi, K. Well-posedness and scattering for the Zakharov system in four dimensions. Analysis & PDE, 8(8), 2029-2055, 2015.

[4] Bejenaru, I., & Herr, S. Convolutions of singular measures and applications to the Za- kharov system. Journal of Functional Analysis, 261(2), 478-506, 2011.

[5] Bejenaru, I., Herr, S., Holmer, J., & Tataru, D. On the 2D Zakharov system with L2 Schrödinger data. Nonlinearity, 22(5), 1063–1089, 2009.

[6] Colliander, James Ellis. ”The initial value problem for the Zakharov system.”, 5427- 5427, 1998.

[7] Chen, T. J., Fang, Y. F., Wang, K. H. Low regularity global well-posedness for the quantum Zakharov system in 1D. Taiwanese Journal of Mathematics, 21(2), 341-36, 2017.

[8] Fang, Y.-F., Shih, H.-W., & Wang, K.-H. Local well-posedness for the quantum Za- kharov system in one spatial dimension. Journal of Hyperbolic Differential Equations, 14(01), 157–192, 2017.

[9] Garcia, L. G., Haas, F., De Oliveira, L.P.L., & Goedert, J. Modified Zakharov equations for plasmas with a quantum correction. Physics of Plasmas, 12(1), 012302, 2005.

[10] Ginibre, Jean, Yoshio Tsutsumi, and Giorgio Velo. ”On the Cauchy problem for the Zakharov system.” Journal of Functional Analysis 151.2, 384-436, 1997.

[11] Guo, Yanfeng, Jingjun Zhang, and Boling Guo. ”Global well-posedness and the classical limit of the solution for the quantum Zakharov system.” Zeitschrift für angewandte Mathematik und Physik 64.1, 53-68, 2013.

[12] Jiang, J. C., Lin, C. K., Shao, S. On one dimensional quantum Zakharov system. arXiv preprint arXiv:1412.2882, 2014.

[13] Ozawa, T., Tsutsumi, Y. Existence and smoothing effect of solutions for the Zakharov equations. Publications of the Research Institute for Mathematical Sciences, 28(3), 329- 361, 1992.

[14] Sulem, C., &PL, S. Quelques résultats de régularité pour les équations de la turbulence de Langmuir, 1979.

[15] Wang, Kuan Hsiang.”On Quantum Zakharov System in One Spatial Dimension.”,1-95, 2017.

[16] Zakharov, Vladimir E.”Collapse of Langmuir waves.” Sov. Phys. JETP 35.5, 908-914, 1972.
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