Constructing characteristic initial data for three dimensional compressible Euler equations

TL;DR

This paper develops a method to construct smooth characteristic initial data for the 3D compressible Euler equations, solving a longstanding open problem and enabling better analysis of flow dynamics.

Contribution

It introduces a recursive vector field method to determine all derivatives of initial data on characteristic surfaces, advancing the understanding of initial value problems in compressible flow.

Findings

Constructed smooth initial data for characteristic surfaces in 3D Euler equations.

Established a recursive method using transport and wave equations for data determination.

Provided tools for analyzing long-time behavior of compressible Euler flows.

Abstract

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.