內容大鋼
克里格編著的《臨界波映射的集中緊性》是一部研究非線性色散方程，特別是幾何發展方程的專著。波映射是在黎曼流形（M，g）上取值的最簡單的波方程，其拉格朗日運算元同標量方程中的基本一樣，僅有的不同是長度的測量與度量g有關。通過Noether定理，拉格朗日對稱表明了波映射的守恆律，如能量守恆。在坐標系中，波映射有半線性系統波方程給出。在過去的20年中，一些表述這個系統的局部和全局適定性問題的重要方法出現了。由於弱色散效應，波映射定義在低維Minkowski空間，如Rt，x1+2上，呈現出特別的技術難題。這一類波函數有格外重要臨界能量特性，事實上即能量尺度和方程極其相似。本書將在雙曲平面中實現集中緊性方法的應用，這一實現的最大挑戰是，將產生更多有關解的詳細信息。
目錄
1 Introduction and overview
1.1 The main result and its history
1.2 Wave maps to H2
1.3 The small data theory
1.4 The Bahouri-G6rard concentration compactness method
1.5 The Kenig-Merle agument
1.6 An overview of the book
2 The spaces S [k] and N [k]
2.1 Preliminaries
2.2 The null-frame spaces
2.3 The energy estimate
2.4 A stronger S[k]-norm, and time localizations
2.5 Solving the inhomogeneous wave equation in the Coulomb gauge
3 Hodge decomposition and null-structures
4 Bilinear estimates involving S and N spaces
4.1 Basic L2-bounds
4.2 An algebra estimate for S[k]
4.3 Bilinear estimates involving both S [kl] and N [k2] waves
4.4 Null-form bounds in the high-high case
4.5 Null-form bounds in the low-high and high-low cases
5 Trilinear estimates
5.1 Reduction to the hyperbolic case
5.2 Trilinear estimates for hyperbolic S-waves
5.3 Improved trilinear estimates with angular alignment
6 Quintilinear and higher nonlinearities
6.1 Error terms of order higher than five
7 Some basic perturbative results
7.1 A blow-up criterion
7.2 Control of wave maps via a fixed L2-profile
8 BMO, Ap, and weighted commutator estimates
9 The Bahouri-Gerard concentration compactness method
9.1 The precise setup for the B ahouri-G6rard method
9.2 Step 1: Frequency decomposition of initial data
9.3 Step 2: Frequency localized approximations to the data
9.4 Step 3: Evolving the lowest-frequency nonatomic part
9.5 Completion of some proofs
9.6 Step 4: Adding the first large component
9.7 Step 5: Invoking the induction hypothesis
9.8 Completion of proofs
9.9 Step 6 of the Bahouri-Gerard process; adding all atoms
10 The proof of the main theorem
10.1 Some preliminary properties of the limiting profiles
10.2 Rigidity I: Harmonic maps and reduction to the self-similar case
10.3 Rigidity II: The self-similar case
1l Appendix
11.1 Completing a proof
11.2 Completion of proofs
11.3 Completion of a proof
11.4 Completion of a proof
11.5 Competion of proofs
References
Index
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