Mathematical Sciences: Nonlinear Wave Equations

这一数学研究的基本主题是分析非线性偏微分方程的解，特别是在数学物理和流体动力学中出现的保守进化方程。重点将放在研究一类方程的存在性问题，特别是周期解的存在性，以及研究演化的先验规律性和其他性质。这项工作仅限于三个目标。首先，在无限维动力系统和KAM理论领域，将继续研究将应用于波动方程的哈密顿摄动技术扩展到其他方程，并研究与更高空间维度研究相关的方法。第二个方向涉及色散方程的平滑特性。人们早就知道，色散演化方程的解可以比它们的初始条件更光滑。但它们的二次范数可以在空间方向上无界增长而不受截止限制。将做一些工作来分离方程中对增长有贡献的部分，并试图对规范大小进行估计。项目的最后一部分研究自由表面和界面中的波。这个问题被称为斯托克斯猜想，涉及到极值形式的孤立波。除了波峰外，这种波都是解析波。本研究的目的是证明：(1)波峰处必须出现一个内角为120度的Lipschitz奇点；(2)极值波剖面是单调递减的，两侧是凸的。偏微分方程是物理科学中数学建模的基础。涉及连续变化的现象，例如在运动、材料和能量中看到的现象，都遵循某些可以用偏导数之间的相互作用和关系来表示的一般规律。数学的关键作用不是说明关系，而是从中提取定性和定量的意义。

The underlying theme of this mathematical research is the analysis of solutions of nonlinear partial differential equations, particularly in conservative evolution equations that arise in mathematical physics and in fluid dynamics. Emphasis will be on the study of the existence question for classes of equations, in particular, the existence of periodic solutions, and the study of a priori regularity and other properties of the evolution. The work is limited to three objectives. First, in the area of infinite dimensional dynamical systems and KAM theory, continuing research will be done in extending techniques of Hamiltonian perturbation applied to the wave equation to other equations and to investigate methods related to studies in higher space dimensions. The second direction involves smoothing properties of dispersive equations. It has been known for some time that solutions of dispersive evolution equations can be smoother than their initial conditions. But their quadratic norms can grow unbounded without cut-off restrictions in spatial directions. Work will be done to isolate the parts of the equations which contribute to the growth and try to develop estimates on the norm size. The final part of the project studies waves in free surfaces and interfaces. The problem, known as Stokes conjecture, concerns the solitary wave of extremal form. Such waves are known to be analytic except at their crest. It is the object of this study to show (i) that a Lipschitz singularity of interior angle 120 degrees must occur at the crest and (ii) that the extremal wave profile is monotone decreasing and convex on either side. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.