Structurally stable singularities and Lipschitz stable optimal transport metrics for the compressible Euler equations
Abstract
It is well known that solutions to the compressible Euler equations can develop singularities in finite time. In this paper, we carry out a detailed analysis on behaviors of solutions up to the time of the first singularity for the one-dimensional compressible Euler equations with general smooth initial data. Our main results consist of three parts. First, for an open dense set of C3 initial data, we show that the solution of Euler equations is twice continuously differentiable except at most finitely many points when the first singularity happens, using Thom's Transversality Theorem. Second, for any initial data in the open dense set of C3 functions given in the first result, we provide the precise asymptotic description of the solution in a semi-neighborhood in the (x,t)-plane of each singular point at the time of the first singularity, and verify that the solution has a cusp-type singularity with Hölder exponent 1/3 at each singular point. The proofs of the first two results are based on the representation of the solution in terms of a semilinear system. Third, for smooth initial data with small BV norm, we construct two Finsler type optimal transport metrics, then under these metrics show that the solution depends Lipschitz continuously on the initial data up to the time of the first singularity, with uniformly bounded Lipschitz constants. In particular, the C1/3 generic singularity is stable in this sense. On the other hand, since our first two results hold for an open and dense set of initial data, any Hölder continuous cusp singularity with exponent other than 1/3 is unstable under initial perturbations.
Related
- Regularity of Structurally Stable Cusp Singularities for Two Families of Quasilinear Wave-type Equations
- Formation of quasi-singularities of shock and implosion type for compressible Euler flows
- Optimal decay rates to the contact wave for 1-D compressible Navier-Stokes equations
- Wave Equations with Lipschitz Nonlinearity of Combined Power and Logarithmic Type: Existence of Global Solutions
- Stochastic heat equations with non-Lipschitz coefficients driven by symmetric α-stable Lévy white noise
BibTeX
@misc{chen2026structurally,
title = {{Structurally stable singularities and Lipschitz stable optimal transport metrics for the compressible Euler equations}},
author = {Geng Chen and Yanbo Hu and Yannan Shen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13943},
eprint = {2609.13943},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13943},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Structurally stable singularities and Lipschitz stable optimal transport metrics for the compressible Euler equations
AU - Geng Chen
AU - Yanbo Hu
AU - Yannan Shen
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.13943
UR - https://arxiv.org/abs/2609.13943
AB - It is well known that solutions to the compressible Euler equations can develop singularities in finite time. In this paper, we carry out a detailed analysis on behaviors of solutions up to the time of the first singularity for the one-dimensional compressible Euler equations with general smooth initial data. Our main results consist of three parts. First, for an open dense set of C^3 initial data, we show that the solution of Euler equations is twice continuously differentiable except at most finitely many points when the first singularity happens, using Thom's Transversality Theorem. Second, for any initial data in the open dense set of C^3 functions given in the first result, we provide the precise asymptotic description of the solution in a semi-neighborhood in the (x,t)-plane of each singular point at the time of the first singularity, and verify that the solution has a cusp-type singularity with Hölder exponent 1/3 at each singular point. The proofs of the first two results are based on the representation of the solution in terms of a semilinear system. Third, for smooth initial data with small BV norm, we construct two Finsler type optimal transport metrics, then under these metrics show that the solution depends Lipschitz continuously on the initial data up to the time of the first singularity, with uniformly bounded Lipschitz constants. In particular, the C^1/3 generic singularity is stable in this sense. On the other hand, since our first two results hold for an open and dense set of initial data, any Hölder continuous cusp singularity with exponent other than 1/3 is unstable under initial perturbations.
ER -