双曲几何流经典解的生命跨度

发布时间：2018-05-07 08:03

  本文选题：双曲几何流 + 柯西问题　； 参考：《浙江大学》2016年博士论文

【摘要】：本文主要针对高维流形上的两类双曲几何流柯西问题的经典解的生命跨度进行了研究。第一章介绍了本文所研究问题的背景、意义及现状等。第二章主要研究黎曼面上的标准的双曲几何流。通过构造逼近解和特征线方法及Hormander的破裂定理,我们得到黎曼面上的标准双曲几何流的径向经典解一定会在有限时间内产生破裂,而且我们对该经典解的生命跨度给出了一个精确估计。第三章主要考虑了多维的标准双曲几何流方程经典解的存在性。我们通过标准的连续性方法,给出了小初值的多维双曲几何流方程经典解的生命跨度的下界估计。第四章主要研究黎曼面上的带耗散项的双曲几何流。我们得到了一个新的方程并利用能量的方法得到了该方程小初值柯西问题的经典解的整体存在性。而且,如果该方程的初值满足适当的假设条件,我们不仅说明了其经典解的整体存在性,还得到了解随着时间趋于无穷时的渐近形态。最后,在附录A我们介绍了双曲Yamabe问题。我们主要针对(1+n)-维的闵氏空间的Yamabe问题解的整体存在性进行了研究。更精确的说,当n≤3时,我们证明了解的整体存在及破裂性,并说明了(1+n)-维的闵氏空间可以共形于某一个具有常数量曲率的时空。同时,当n≥4时,我们考虑了双曲Yamabe问题的一类特解,并分析了解的存在性。
[Abstract]:In this paper, the life span of the classical solution of two classes of hyperbolic geometric flow Cauchy problem on high dimensional manifold is studied. The first chapter introduces the background, significance and current situation of the problems studied in this paper. In the second chapter, the standard hyperbolic geometric flow on Riemannian surface is studied. By constructing approximate solution, characteristic line method and Hormander's rupture theorem, we obtain that the radial classical solution of standard hyperbolic geometric flow on Riemannian surface must produce rupture in finite time. Moreover, we give an exact estimate of the life span of the classical solution. In chapter 3, we consider the existence of classical solutions for multidimensional standard hyperbolic geometric flow equations. By using the standard continuity method, we give the lower bound estimates of the life span of the classical solutions of the multi-dimensional hyperbolic geometric flow equations with small initial values. In chapter 4, the hyperbolic geometric flow with dissipative term on Riemannian surface is studied. We obtain a new equation and obtain the global existence of the classical solution of the Cauchy problem with small initial value by using the energy method. Furthermore, if the initial value of the equation satisfies the proper assumptions, we not only show the global existence of its classical solution, but also obtain the asymptotic behavior of the solution as time approaches infinity. Finally, we introduce the hyperbolic Yamabe problem in Appendix A. In this paper, we study the global existence of solutions to the Yamabe problem in a 1-nm-dimensional Mindahl space. More precisely, when n 鈮，

本文编号：1856103