Mathematical Sciences: Nonlinear Wave Equations

这项数学研究的基本主题是 非线性偏微分方程解的分析 方程，特别是保守的演化方程， 数学物理学和流体动力学中出现的。 重点 将研究的存在性问题的类 方程，特别是，周期解的存在性， 和研究的先验规律性和其他性质的 进化 这项工作限于三个目标。 一是在 无穷维动力系统领域与KAM 理论，继续研究将在扩展技术 哈密顿扰动应用于波动方程的其他 方程，并探讨与高等教育研究有关的方法， 空间维度 第二个方向涉及平滑 色散方程的性质 它已经被称为一些 色散演化方程的解可以 比初始条件更好。 但他们的二次方程式 规范可以无限增长，在空间上没有截止限制， 方向 工作将被做隔离的部分， 这些方程有助于增长并试图发展 对标准尺寸的估计。 项目的最后一部分 研究自由表面和界面中的波。 问题是， 称为斯托克斯猜想，涉及孤立波的 极值形式 已知这种波是解析波， 他们的Crest。 本研究的目的是表明（i） 内角为120度的Lipschitz奇点必须出现在 波峰和（ii）极值波形是单调的 在两侧逐渐减小和凸起。 偏微分方程是 物理科学中的数学建模。 的现象 包括连续变化，如运动、材料 和能量都服从某些普遍规律，这些规律可以用 部分之间的相互作用和关系的术语 衍生物. 数学的关键作用不是陈述 关系，而是要提取定性和 从数量意义上来说。

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.