【作者】 刘磊；

【导师】 李嘉禹；

【作者基本信息】 中国科学技术大学 ， 基础数学， 2014， 博士

【摘要】 爆破分析是偏微分方程和几何分析研究中的一个重要课题,其主要包括能量等式和no neck性质。本文的主要研究对象是调和映射及调和映射流的一些推广,例如Dirac-调和映射、双调和映射以及双调和映射流。本文证明了Dirac-调和映射和双调和映射的no neck性质。此外,本文还构造出双调和映射流在有限时间爆破的一些例子。具体而言,本文的主要结果列举如下。I.Dirac-调和映射的neck分析定理0.1.设(M,9)为紧的黎曼曲面,∑M是M上的spin从,(N,h)为任一紧的黎曼流形,{Φk,ψk}是从(M,g)到(N,h)的一列能量有界的Dirac-调和映射,设{Φk,ψk}弱收敛到Dirac-调和映射{Φ,ψ},P1,…,PI是爆破点集合,那么存在子列(不妨仍记为{Φk,ψk})和有限个Dirac-调和映射球{σli,ξli},i=1,...,I;l=1,...,Lt,使得Φ(D)∪ii1∪Lil=1(σli(S2))是连通的。Ⅱ.双调和映射的neck分析定理0.2.设B1是R4中的单位球,(N,h)是黎曼流形,ui是从B1到N的一列双调和映射满足其中A是常数。假设存在一列正数λi→0使得对于任意R4中的紧集K,有若ω是唯一的bubble,u∞是ui的弱极限,则我们有注记0.3.为了叙述上简单起见,在上述定理中我们做了一个不必要的假设,即在爆破过程中只有一个bubble。事实上,根据Ding和Tian[15]的归纳假设我们知道定理对于多个bubble的情形同样是成立的。利用上述定理的证明方法我们还可以给出双调和映射的能量等式和可去奇点的新证明。即我们有以下两个推论：推论0.4.在上定理的假设条件下,我们有推论0.5.设u是定义在B1\{0}上的光滑双调和映射,如果则u可以光滑的延拓到B1。Ⅲ.双调和映射流的有限时间爆破定理0.6.设uit是从(B1,9)到(N,h)的一列逼近双调和映射,即ut满足其中9是球面度量。设A>0和p≥4/3为两常数,且假设存在一列正数λi→0使得对于任意R4中的紧集K,都有强收敛若ω是唯一的bubble,u∞是ui的弱极限,则我们有定理0.7.设M’是一维数大于4的闭的流形且π4(M’)≠(?),记Tm是m维环面,M=M’#Tm(连通和)。对于M上的任何度量9,我们都可以找到初始映射u00：S4→M使得从u0出发的双调和映射流在有限时间爆破。更多还原

【Abstract】 Blow-up analysis is an important topic in the study of partial differential equations and geometric analysis, which usually include energy identity and neck analysis. In the present paper, we are concerned on the popularization of harmonic maps and harmonic map flow, such as Dirac-harmonic maps, biharmonic maps and biharmonic map flow. I will prove in this paper that there is no neck during the blow-up process of Dirac-harmonic maps and biharmonic maps. On the other hand, I will construct some initial maps and the target manifold with some topology condition to prove the biharmonic map flow must blowup in finite time. In detail, the main results of the paper are as follows.Ⅰ.Neck analysis of Dirac-harmonic mapsTheorem0.1. Let (M, g) be a compact Riemannian surface and SM be the spin bun-dle over M.(N, h) be another compact Riemannian manifold. For a sequence of smooth Dirac-harmonic maps {(φκ,ψκ)} with uniform bounded energy E(φκ,ψκ）≤∧<A∞, we assume that{(φκ,ψκ)} weakly converges to a Dirac-harmonic map{(φ,ψ)} in W1,2(M, N)×L4(∑M(?)RK) with a finite blow-up points {pl,..., pI}, then after taking subsequence, still denoted by{(φκ,ψκ)}, we can find a finite set of Dirac-harmonic spheres (σil,fεil):S2→N,i=1,....I;l=1,...,Li such that the image φ(D) UiI=1UlLi=1(σil(S2)) is a connected set.Ⅱ.Neck analysis of biharmonic mapsTheorem0.2. Suppose B4(?) R4is a unite sphere and (N, h) is a compact Riemannian manifold. Let Ui be a sequence of biharmonic maps from B4to N satisfying for some A>O. Assume that there is a sequence positive λi→O such that ui(λix)→w on any compact set K K (?) R4, that ui converges weakly in W2,2to u∞and that w is the only bubble. Then,Remark0.3. For convenient, we make a unnecessary assumption in the above theorem, that is there is only one bubble in the blow-up process. But, according to Ding and Tian[15I’s induction method, we know the above theorem also holds for two or more bubbles.By the proof of above theorem, I can give a new method to prove the energy identity and removable singularity of biharmonic maps. So, we can get the following two corollary.Corollary0.4. Under the same assumption of above theorem, we haveCorollary0.5. Let u be a smooth biharmonic map on B1\{0}. If then u can be extended to a smooth biharmonic map on B1.Ⅲ.Finite time blow-up of biharmonic map flowTheorem0.6. Let ui be a sequence of approximate biharmonic maps from B4to (N, h) satisfying with for some∧>0and p≥4/3. Assume that there is a positive sequence λi→0such that on any compact set K C (?) R4, that ui converges weakly in W2,2to u∞and that w is the only bubble. Then, Theorem0.7. Suppose that M’is any closed manifold of dimension m>4with nontrivial π4(M’) and let M’##Tm be the connected sum of M’ with the torus of the same dimension. For any Riemannian metric g on M, we can find (infinitely many) initial map u0:S4→M such that the biharmonic map flow starting from u0develops finite time singularity.更多还原