Partial Differential Equations

本文主要研究了偏微分方程中具有理论意义和实际意义的几个问题：第一部分是关于平方和算子解的正则性问题，在满足括号条件的情况下，讨论了解的局部正则性和整体正则性的充要条件，并给出了解的正则性的一个证明。在没有括号条件的情况下，我们将研究环面上的次拉普拉斯算子的整体解析正则性和光滑正则性的较弱性质，并考虑CR结构的正则性问题，第二部分研究非线性浅水型偏微分方程在低正则初值下的Cauchy问题，谐波分析技术特别地，我们计划研究完全可积Camassa-Holm方程初值问题的局部和整体适定性，最后，在第三部分中，我们计划证明紧黎曼流形的保体积同态群上的适当度量的指数映射是指数为零的非线性Fredholm映射。这是很重要的，因为众所周知，这个度规的测地线对应于流体力学的欧拉方程的解。偏微分方程是一个多方面的学科。我们对自然界基本过程的理解在很大程度上是基于偏微分方程的。例如，在我们建议的第一部分中考虑的方程产生于化学品的扩散，热的传播，在许多其它物理过程中，扩散都是受扩散影响的，而且扩散过程与随机（布朗）运动密切相关。这些扩散偏微分方程理论与概率论的综合为研究存在扩散和随机性的问题提供了理论框架。数学和物理学之外的一个例子是金融领域，在那里，来自随机分析的强大技术已经被应用于几乎所有的金融数学方面：金融衍生产品的定价，如期权和债券，套期保值，利率等等。第二部分和第三部分的方程来自于流体力学问题。它们是描述流体流动的数学模型。对这些方程的研究将有助于我们理解波的形成、奇点的结构以及它们的长时间行为。同时，研究这些偏微分方程问题的理论形成了一个庞大的学科，它与许多其他数学分支相互作用，如复分析、微分几何、调和分析、概率论和数学物理。

DMS-9970857ABSTRACTOur mathematical research centers around the followingproblems in partial differential equations of boththeoretical and practical interest.The first part is concerned with the regularity of solutionsfor the sum of squares operator, or sublaplacian.We will investigate necessary andsufficient conditions for the local and global analytic regularity of thesublaplacian when the bracket condition is satisfied.Moreover, in the absence of the bracket condition we will studythe weaker properties of global analytic and smooth regularityof the sublaplacian on the torus.Also we will consider regularity problems for CR structures,and principal type operators.The second part is concerned with the studyof the Cauchy problem for nonlinear partial differential equationsof shallow water type under low regularity initial datausing harmonic analysis techniques. In particular weplan to investigate the local and global well-posedness of theinitial value problem for the completely integrableCamassa-Holm equation.And finally, in the third part we plan to show that the exponential map ofan appropriate metric on the group of volume--preserving diffeomorphisms of acompact riemannian manifold is a nonlinear Fredholm map ofindex zero. This would be significant because, as is well known,geodesics of this metric correspond to solutions of the Eulerequations of hydrodynamics.Partial differential equations is a many-faceted subject.Our understanding of the fundamental processes of the natural worldis based to a large extent on partial differential equations.For example, the equations considered in the first part of ourproposal arise in the diffusion of chemicals, in the spread of heat,and in many other physical processes influenced by diffusion.Moreover diffusion processes are closely connected torandom (Brownian) motions. The synthesis of the theory of thesediffusion partial differential equations and probability provides the theoretical framework for studying problemswhere diffusion and randomness are present. An example outsidemathematics and physics is the field of Finance, where powerful techniques fromstochastic analysis have been brought to bear on almost all aspects ofmathematical finance: pricing of financial derivative products suchas options and bonds, hedging, interest rates and so on.The equations in the second and third parts arise from problems inhydrodynamics. They are mathematical models describingfluid flow. The study of these equations will contributein our understanding of wave formation, the structure of theirsingularities, andtheir long time behavior. It may also contribute in the verybig problem of understanding turbulence.At the same time the theory for studying these partial differentialequations problems forms a vast subject that interacts with many otherbranchesof mathematics, such as complex analysis, differential geometry, harmonicanalysis, probability, and mathematical physics.