标题： 构造三维可压缩欧拉方程的特征初始数据 显示英文标题

标题： Constructing characteristic initial data for three dimensional compressible Euler equations

摘要： 本文解决了三维可压缩欧拉方程的特征初始数据问题——这是一个与广义相对论中真空爱因斯坦场方程的Christodoulou特征初值公式类似的一个开放问题。 在声学几何的框架下，我们证明了对于任何“初始锥”$C_0\subset \mathcal{D}=[0,T]\times\mathbb{R}^3$，在$S_{0,0}=C_0\cap \Sigma_0$处给出初始数据$(\mathring{\rho},\mathring{v},\mathring{s})$，任意光滑的熵函数和角速度确定在$C_0$上的光滑初始数据$(\rho,v,s)$，使得$C_0$为特征的。 与Speck-Yu [19] 的相交超曲面情况和Lisibach [11] 的对称约化情况不同，我们的向量场方法通过传输方程和波动方程递归地确定解沿$C_0$的所有（包括$0$-阶）导数。 这项工作为三维可压缩欧拉系统中允许的超曲面提供了完整的特征数据构造，引入了有用的工具，并为可压缩欧拉流的长时间动力学研究提供了新的视角。 显示英文摘要

摘要： This paper resolves the characteristic initial data problem for the three-dimensional compressible Euler equations - an open problem analogous to Christodoulou's characteristic initial value formulation for the vacuum Einstein field equations in general relativity. Within the framework of acoustical geometry, we prove that for any "initial cone" $C_0\subset \mathcal{D}=[0,T]\times\mathbb{R}^3$ with initial data $(\mathring{\rho},\mathring{v},\mathring{s})$ given at $S_{0,0}=C_0\cap \Sigma_0$, arbitrary smooth entropy function and angular velocity determine smooth initial data $(\rho,v,s)$ on $C_0$ that render $C_0$ characteristic. Differing from the intersecting-hypersurface case by Speck-Yu [19] and the symmetric reduction case by Lisibach [11], our vector field method recursively determines all (including $0$-th) order derivatives of the solution along $C_0$ via transport equations and wave equations. This work provides a complete characteristic data construction for admissible hypersurfaces in the 3D compressible Euler system, introducing useful tools and providing novel aspects for studies of the long-time dynamics of the compressible Euler flow.