在這篇論文裡，我們研究一些關於一維量子Zakharov 系統的議題。這些議題涉及局部適定性、不適定性、全局適定性，還有半古典極限問題。另外，對於四階薛丁格方程和四階波方程的解我們獲得了Strichartz 式的估計。一些議題對於Zakharov 系統已經有廣泛的研究了，一些方法與技巧被Ginibre、Tsutsumi、Velo [14]，Holmer [18]，Colliander、Holmer、Tzirakis [6] 和Guo、Zhang、Guo [16] 應用在Zakharov 系統上，對於量子Zakharov 系統，我們適當地修改這些方法與技巧去獲得新的結果在相對應的議題上。
在量子Zakharov 系統的局部適定性上，我們得到了一個比Jiang、Lin、Shao [20]更大的範圍，然後我們在局部適定性的範圍外面建立了兩個關於不適定性的結果。緊接著，我們證明量子Zakharov 系統的全局適定性當薛丁格的初始資料在L2 空間，當量子參數趨近於0 時，我們在形式上獲得了Colliander、Holmer、Tzirakis 在Zakharov系統上的結果。我們也考慮量子Zakharov 系統與Zakharov 系統的半古典極限問題並且改進了Guo、Zhang、Guo [16] 的結果。

In this dissertation, we research some topics about the quantum Zakharov system in one spatial dimension and the topics including local well-posedness, ill-posedness, global wellposedness, and semi-classical limit problems. In addition, we get the Strichartz estimates for the solutions of fourth-order Schrödinger equation and fourth-order wave equation. Some topics have been extensively studied for Zakharov system. To obtain new results of related topics for the quantum Zakharov system, we appropriately modify some methods and techniques which applied to the Zakharov system by Ginibre, Tsutsumi, and Velo [14], Holmer [18], Colliander, Holmer, and Tzirakis [6], and Guo, Zhang, and Guo [16].
For the local well-posedness of quantum Zakharov system, we get a larger region than Jiang, Lin, and Shao [20]. Then we establish two results about the ill- posedness of quantum Zakharov system outside of the local well-posedness region. Next, we prove the global well-posedness of quantum Zakharov system with L2-Schrödinger data. As the quantum parameter tends to zero, we formally get the result of Colliander, Holmer, and Tzirakis [6] for the Zakharov system. We also consider the semi-classical limit for the quantum Zakharov system and the Zakharov system, and improve the result of Guo, Zhang, and Guo [16].

[1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3(1993), no. 2, 107-156.
[2] J. Bourgain and J. Colliander, On well-posedness of the Zakharov system. Int. Math. Res. Not. 1996(1996), 515-546.
[3] I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system. J. Funct. Anal. 261(2011), 478-506.
[4] I. Bejenaru, S. Herr, J. Holmer, and D. Tataru, On the 2D Zakharov system with L2- Schrödinger data. Nonlinearity 22(2009), 1063–1089.
[5] 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), 341-361.
[6] J. Colliander, J. Holmer, and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. AMS. 360(2008), 4619-4638.
[7] Y.F. Fang, C.K. Lin and J.I. Segata, The fourth-order nonlinear Schrödinger limit for quantum Zakharov system. Z. Angew. Math. Phys. (2016), 67:145.
[8] Y.F. Fang, H.W. Shih, and K.H. Wang, Local well-posedness for the quantum Zakharov system in one spatial dimension. J. Hyper. Differential Equations. 14(2017), 157-192.
[9] 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 012302 (2005).
[10] L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two. I. Comm. Math. Phys. 160(1994), 173-215.
[11] L. Glangetas and F. Merle, Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two. II. Comm. Math. Phys. 160(1994), 349-389.
[12] 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.
[13] Z. Guo, L. Peng, and B. Wang, Decay estimates for a class of wave equations. J. Funct. Anal. 254(2008), 1642-1660.
[14] J. Ginibre, Y. Tsutsumi, and G. Velo, On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151(1997), 384-436.
[15] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133(1995), 50-68.
[16] 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.
[17] F. Haas,“Quantum plasmas”. Springer Series on Atomics, Optical and Plasma Physics 65(2011).
[18] J. Holmer, Local ill-posedness of the 1D Zakharov System. Electronic JDE 2007(2007) no. 24, pp. 1-22.
[19] F. Haas and P.K. Shukla, Quantum and classical dynamics of Langmuir wave packets. Rev. E 79 066402 (2009).
[20] J.C. Jiang, C.K. Lin, and S. Shao, On One Dimensional Quantum Zakharov System, Discrete and Continuous Dynamical System, Series A, 36(2016), pp. 5445-5475.
[21] C. Kenig, G. Ponce, and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69.
[22] C. Kenig, G. Ponce, and L. Vega, A bilinear estimate eith applications to the KdV equation. J. Amer. Math. Soc. 9(1996), 573-603.
[23] M. Keel and T. Tao, Endpoint Strichartz estimates, American Journal of Mathematics 120(1998), 955-980.
[24] N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems. Invent. Math. 172(2008), 535-583.
[25] T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations. Differential Integral Equations 5(1992), 721-745.
[26] T. Ozawa and Y. Tsutsumi, Global existence and asymptotic behavior of solutions for the Zakharov equations in three space dimensions. Adv. Math. Sci. Appl. 3(1993/94), Special Issue, 301-334.
[27] B. Pausader, Global Well-Posedness for Energy Critical Fourth-Order Schrödinger Equations in the Radial Case. Dynamics of PDE 4, No.3, (2007), 197-225.
[28] H. Pecher, Global well-posedness below energy space for the 1-dimensional Zakharov system. Internat. Math. Res. Notices (2001), no. 19, 1027-1056. MR1857386 (2002j:35036)
[29] H. Pecher, Global solutions with infinite energy for the one-dimensional Zakharov system. Electron. J. Differential Equations (2005), No. 41, 18 pp.(electronic). MR2135252
[30] R.S.Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equation. Duke Math. J. 44(1977), 705-714.
[31] S.H. Schochet and M.I. Weinstein,The nonlinear Schrödinger limit of the Zakharov equations govrning Langmuir turbulence. Commun. Math. Phys. 106(1986), 569-580.
[32] V.E. Zakharov, Collapse of Langmuir waves. Sov. Phys. JETP 35 (1972), 908-914.