本文選題：雙曲幾何流 + 柯西問(wèn)題　； 參考：《浙江大學(xué)》2016年博士論文

【摘要】：本文主要針對(duì)高維流形上的兩類(lèi)雙曲幾何流柯西問(wèn)題的經(jīng)典解的生命跨度進(jìn)行了研究。第一章介紹了本文所研究問(wèn)題的背景、意義及現(xiàn)狀等。第二章主要研究黎曼面上的標(biāo)準(zhǔn)的雙曲幾何流。通過(guò)構(gòu)造逼近解和特征線(xiàn)方法及Hormander的破裂定理,我們得到黎曼面上的標(biāo)準(zhǔn)雙曲幾何流的徑向經(jīng)典解一定會(huì)在有限時(shí)間內(nèi)產(chǎn)生破裂,而且我們對(duì)該經(jīng)典解的生命跨度給出了一個(gè)精確估計(jì)。第三章主要考慮了多維的標(biāo)準(zhǔn)雙曲幾何流方程經(jīng)典解的存在性。我們通過(guò)標(biāo)準(zhǔn)的連續(xù)性方法,給出了小初值的多維雙曲幾何流方程經(jīng)典解的生命跨度的下界估計(jì)。第四章主要研究黎曼面上的帶耗散項(xiàng)的雙曲幾何流。我們得到了一個(gè)新的方程并利用能量的方法得到了該方程小初值柯西問(wèn)題的經(jīng)典解的整體存在性。而且,如果該方程的初值滿(mǎn)足適當(dāng)?shù)募僭O(shè)條件,我們不僅說(shuō)明了其經(jīng)典解的整體存在性,還得到了解隨著時(shí)間趨于無(wú)窮時(shí)的漸近形態(tài)。最后,在附錄A我們介紹了雙曲Yamabe問(wèn)題。我們主要針對(duì)(1+n)-維的閔氏空間的Yamabe問(wèn)題解的整體存在性進(jìn)行了研究。更精確的說(shuō),當(dāng)n≤3時(shí),我們證明了解的整體存在及破裂性,并說(shuō)明了(1+n)-維的閔氏空間可以共形于某一個(gè)具有常數(shù)量曲率的時(shí)空。同時(shí),當(dāng)n≥4時(shí),我們考慮了雙曲Yamabe問(wèn)題的一類(lèi)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，

本文編號(hào)：1856103