The Characteristic Initial Value Problem for the 3D Compressible Euler Equations
Abstract
We prove local existence and uniqueness, propagation of Sobolev regularity, and continuation criteria results for the characteristic initial value problem for the 3D compressible Euler equations. This is the Cauchy problem with initial data given on a pair of transversally intersecting (acoustically) null hypersurfaces. The region of existence in our main theorem is a future-neighborhood of the union of the data null hypersurfaces, where the size of the neighborhood depends on various norms of the data as well as the structure of the data null hypersurfaces. We allow for any equation of state with positive sound speed, and we do not make any symmetry, irrotationality, or isentropicity assumptions. Our analysis relies on the geometric formulation of the flow derived in [Speck19], which decomposes the equations into a "wave-part" coupled to a "transport-div-curl" part. Compared to Rendall's celebrated solution of the characteristic initial value problem for wave equations [Rendall90], substantial new ideas are needed to handle the fact that the compressible Euler equations are more complicated than pure wave equations.
Related
- Picard Iteration for the Characteristic Initial Value Problem in Einstein Equations
- Finite-Time Vanishing of Vacuum and Large-Time Behavior of the Multi-Dimensional Degenerate Compressible Navier-Stokes Equations with Large Spherically Symmetric Initial Data
- Large time behavior of global solutions to one-dimensional non-isentropic compressible Navier-Stokes-Poisson equations with large initial data
- Shock solutions of a 3D spherical piston problem for the compressible Euler equations
- Sharp sign criterion for global existence and blow-up of strong solutions with arbitrarily large initial data in the 3D spherically symmetric compressible Navier-Stokes equations
BibTeX
@misc{speck2026characteristic,
title = {{The Characteristic Initial Value Problem for the 3D Compressible Euler Equations}},
author = {Jared Speck and Sifan Yu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04201},
eprint = {2611.04201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04201},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Characteristic Initial Value Problem for the 3D Compressible Euler Equations
AU - Jared Speck
AU - Sifan Yu
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04201
UR - https://arxiv.org/abs/2611.04201
AB - We prove local existence and uniqueness, propagation of Sobolev regularity, and continuation criteria results for the characteristic initial value problem for the 3D compressible Euler equations. This is the Cauchy problem with initial data given on a pair of transversally intersecting (acoustically) null hypersurfaces. The region of existence in our main theorem is a future-neighborhood of the union of the data null hypersurfaces, where the size of the neighborhood depends on various norms of the data as well as the structure of the data null hypersurfaces. We allow for any equation of state with positive sound speed, and we do not make any symmetry, irrotationality, or isentropicity assumptions. Our analysis relies on the geometric formulation of the flow derived in [Speck19], which decomposes the equations into a "wave-part" coupled to a "transport-div-curl" part. Compared to Rendall's celebrated solution of the characteristic initial value problem for wave equations [Rendall90], substantial new ideas are needed to handle the fact that the compressible Euler equations are more complicated than pure wave equations.
ER -