摘要
本文利用时空估计方法处理几类非线性方程(组)的适定性问题。众所周知，对于线性方程，一旦基于振荡积分建立起L~p-L~q(L~p-L~(p′))估计，就能利用TT~*方法得到时空估计。在论文的开头，我们简要回顾三类经典方程(Schr(o|¨)dinger方程、波动方程及Klein-Gordon方程)的p-p′估计以及时空估计，它们是进行下一步研究必不可少的工具。
具备Yukawa相互作用的Klein-Gordon-Schr(o|¨)dinger耦合方程组是描述复核子(nucleon)场与实质子(meson)场的相互作用的经典模型，其中u表示复核子场，v表示实质子场，实数μ表示质子的质量。对方程组(0．1)适定性的研究，已有一些结果[2，23，32，63]。在[27]和[28]这两篇文章中，J．Ginibre和G．Velo分别考虑非线性Klein-Gordon方程和非线性Schr(o|¨)dinger方程在能量空间的适定性。他们通过假定非线性项满足适当的基本假设，利用Galerkin方法得到弱解的整体存在性。在非线性项满足更强的基本假设的前提下，利用时空估计以及非线性估计证明了前面得到的弱解是唯一的，这样就得到方程的适定性。受他们的启发，我们萌生了用这种方法处理非线性K-G-S方程组的念头。同时，我们也面临着这样的问题：如何把(0．1)的非线性项推广到一般情形?如何对非线性项附加基本假设才能确保方程组弱解的整体存在性?该方法唯一性的证明过程依赖于时空估计，而两类方程的时空容许对却截然不同，该如何选取合适的时空容许对呢?在[28]中对非线性项的增长指标有一个很不自然的限制条件，我们是否可以去掉这个条件呢?所有的这些问题都随着研究的深入逐一得到回答。
在论文的第一部分，我们考虑非线性K-G-S方程组
In this thesis, we use time-space estimate method to deal with several kinds of nonlinear evolution equation(s). It's well-known that once L~p - L~q(L~p- L~p') estimate is constructed on the basis of oscillation integral, we can obtain time-space estimate by TT~* method for linear equation. Firstly, we recall p - p' estimates and time-space estimates of three classical equations (Schrodinger equation, wave equation and Klein-Gordon equation), which are necessary tools to continue our research.
Klein-Gorden-schrodinger equations
describe a classical model of Yukawa's interaction of conversed complex nucleon field with neutral real meson field. Here u represents a complex scalar nucleon field and v a real scalar meson field. The real constant μ describes the mass of a meson.
For well-posedness of (0.1), one may refer readers to [2, 23, 32, 63]. In 1985, J. Ginibre & G. Velo considered the well-posedness in energy-space of pure Klein-Gordon and pure Schrodinger equations in their papers [27, 28]. Through imposing assumptions on nonlinearity f, they obtained the global existence of weak solutions by Galerkin method. Under stronger assumptions, they obtain the uniqueness of those solutions by usage of Strichartz estimates and estimates for nonlinearity. Inspired by this approach, we beginning to consider K-G-S equations. At the same time, we are confronted with several questions. How to generalize the nonlinearities of (0.1)? How to impose assumptions on nonlinearities to ensure the global existence of weak solutions? The method depends on Strichartz estimate while the admissible pairs of two kinds of equations are different, how to choose the admissible pairs? Can we remove the unnatural restriction on the growth exponent of nonlinearities? All of these questions are to be answered one by one with our further research.
In the first part of this thesis, we consider well-posedness of generalized nonlinear K-G-S
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