某些非线性发展方程的整体解

The quantum Zakharov system is a general model to analyze the coupling between Langmuir waves and ion-acoustic waves in a quantum setting. Contrary to the quantum degenerate case, the present model is more suitable to investigate the classical limit. If we concern its simultaneous semiclassical and static limit, then the quantum nonlinear Schrödinger (NLS) equation can be produced. The quantum NLS equation can be used to study perturbations of the classical NLS soliton solutions. It has important physical and practical significance to study these equations. This project mainly studies the modified equations for plasmas with a quantum correction and correlation equations. We hope to solve the following issues through the classic theories of PDE: the well-posedness of global solutions to quantum Zakharov equation, the asymptotic behavior of the solution to quantum Zakharov equation, the well-posedness of the global solutions to quantum NLS equation. By Galerkin method and a priori estimates, We obtain the existence of global solution. we regard quantum Zakharov equations as Langmuir Turbulence with parameters and study the limit behavior of the systems. Actually, our aim is to confirm and make precise in what sense the solutions of quantum equations converge to the unique solution of Zakharov equation. This project has the frontier and extensive application prospect.

量子Zakharov方程是量子环境中分析朗缪尔波和离子声波之间耦合的通用模型，相对于量子退化的情况，更适合去研究经典的极限过程。若对量子Zakharov方程同时考虑半经典和静态极限，可以得到量子非线性Schrödinger（NLS）方程，这个方程可以用来研究经典NLS方程孤子解的扰动。对这些方程的研究具有重要的物理意义和现实意义。本项目主要研究量子修正的等离子体所满足的方程及其相关方程，拟通过非线性偏微分方程经典理论解决以下问题：量子Zakharov方程整体解的适定性、量子Zakharov方程解的渐近行为、量子NLS方程整体解的适定性。拟用Galerkin方法结合一系列的先验估计获得整体解的存在性。拟将量子Zakharov方程视为带参数的朗缪尔扰动方程以研究解的渐近行为，讨论该方程解的弱极限和强极限，确定在什么意义下该方程的解收敛到Zakharov方程的唯一解。本课题具有前沿性和应用前景。

本项目研究了来源于现代物理学的一些非线性发展方程的适定性和渐近行为等问题。首先研究了三维量子Zakharov方程的适定性。利用先验估计、Galerkin方法及紧致性原理等得到了三维量子Zakharov方程Cauchy问题整体光滑解的存在性和唯一性。接着研究了三类广义Zakharov方程的适定性。利用能量方法，嵌入定理和一系列精细的插值不等式等，分别得到了相应方程初值问题整体光滑解的存在性和唯一性。最后研究了广义Zakharov方程的渐近行为，将广义Zakharov方程视为带参数的Langmuir扰动方程，研究当参数趋于零时，广义Zakharov方程初值问题解的弱极限和强极限行为。利用非线性偏微分方程的经典理论，得到了不同范数意义下广义Zakharov方程收敛到经典Zakharov方程的结论。