Constructing characteristic initial data for three dimensional compressible Euler equations

2025/08/21 by Yuxuan Wang, Sifan Yu, Wang, Yuxuan +3
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2508.15199

openalex publication_date 2025/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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" C₀⊂ D=[0,T]×ℝ³ with initial data (\mathringρ,\mathringv,\mathrings) given at S_0,0=C₀∩ Σ₀, arbitrary smooth entropy function and angular velocity determine smooth initial data (ρ,v,s) on C₀ that render C₀ 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 C₀ 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.

Citations

• The relativistic Euler equations: ESI notes on their geo-analytic structures and implications for shocks in 1D and multi-dimensions
• The emergence of the singular boundary from the crease in 3D compressible Euler flow
• Shock Reflection in Plane Symmetry
• The Shock Development Problem
• A New Formulation of the 3D Compressible Euler Equations With Dynamic\n Entropy: Remarkable Null Structures and Regularity Properties
• The hidden null structure of the compressible Euler equations and a prelude to applications
• Characteristic Initial Value Problem for Spherically Symmetric Barotropic Flow
• Compressible Flow and Euler's Equations
• On the Local Existence for the Characteristic Initial Value Problem in\n General Relativity
• The Formation of Black Holes in General Relativity

Related

• A New Formulation of the 3D Compressible Euler Equations With Dynamic\n Entropy: Remarkable Null Structures and Regularity Properties
• Rough solutions of the 3-D compressible Euler equations
• Compressible Flow and Euler's Equations
• Well-posedness for rough solutions of the 3D compressible Euler equations
• Remarkable localized integral identities for 3D compressible Euler flow and the double-null framework