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内容简介
本书主要考虑三维空间中，其初值在单位球面外为常值的任意状态方程的经典可压缩欧拉方程。当初值与常状态差别适当小时，我们建立的定理可以给出关于解的完整描述。地，解的定义域的边界包含一个奇异部分，在那里波前的密度将会趋向于无穷大，从而激波形成。在本书中，我们采用几何化方法，得到了关于这个奇异部分的完整的几何描述以及解在这部分性态的详细分析，其核心概念是声学时空流形。
目录
1 Compressible Flow and Non-linear Wave Equations
1.1 Euler's Equations
1.2 Irrotational Flow and the Nonlinear Wave Equation
1.3 The Equation of Variations and the Acoustical Metric
1.4 The Fundamental Variations
2 The Basic Geometric Construction
2.1 Null Foliation Associated with the Acoustical Metric
2.1.1 Galilean Spacetime
2.1.2 Null Foliation and Acoustical Coordinates
2.2 A Geometric Interpretation for Function H
3 The Acoustical Structure Equations
3.1 The Acoustical Structure Equations
3.2 The Derivatives of the Rectangular Components of L and T
4 The Acoustical Curvature
4.1 expressions for Curvature Tensor
4.2 Regularity for the Acoustical Structure Equations as μ → 0
4.3 A Remark
5 The Fundamental Energy Estimate
5.1 Bootstrap Assumptions. Statement of the Theorem
5.2 The Multiplier Fields K0 and K1. The Associated Energy-Momentum Density Vectorfields
5.3 The Error Integrals
5.4 The Estimates for the Error Integrals
5.5 Treatment of the Integral Inequalities Depending on t and u.
Completion of the Proof
6 Construction of Commutation Vectorfields
6.1 Commutation Vectorfields and Their Deformation Tensors
6.2 Preliminary Estimates for the Deformation Tensors
7 Outline of the Derived Estimates of Each Order
7.1 The Inhomogeneous Wave Equations for the Higher Order Variations. The Recursion Formula for the Source Functions
7.2 The First Term in ρn
7.3 The Estimates of the Contribution of the First Term in ρn to the Error Integrals
8 Regularization of the Propagation Equation for □trX. Estimates for the Top Order Angular Derivatives of X
8.1 Preliminary
8.1.1 Regularization of The Propagation Equation
8.1.2 Propagation Equations for Higher Order Angular Derivatives
8.1.3 Elliptic Theory on St,u
8.1.4 Preliminary Estimates for the Solutions of the Propagation Equations
8.2 Crucial Lemmas Concerning the Behavior of μ
8.3 The Actual Estimates for the Solutions of the Propagation Equations
9 Regularization of the Propagation Equation for □μ.
Estimates for the Top Order Spatial Derivatives of μ
9.1 Regularization of the Propagation Equation
9.2 Propagation Equations for the Higher Order Spatial Derivatives
9.3 Elliptic Theory on St,u
9.4 The Estimates for the Solutions of the Propagation Equations
10 Control of the Angular Derivatives of the First Derivatives of
the xi. Assumptions and Estimates in Regard to X
10.1 Preliminary