Formation of quasi-singularities of shock and implosion type for compressible Euler flows
Abstract
This paper investigates a novel quasi-singularity formation phenomenon in the isentropic compressible Euler equations in three dimensions. For any prescribed finite set of points {zj}j=1N ⊂ ℝ3 and an arbitrarily large parameter M ≫ 1, we explicitly construct real-analytic initial data generated by Herglotz-type wave fields. We prove that the corresponding classical solution (p,v)∈ C1([0,T]×ℝ3;ℝ×ℝ3), with T = C0 M−(3γ+1)/(γ−1), where γ> 1 is the adiabatic index and C0 > 0 is a constant independent of M. Throughout the time interval $[0, T]$, the velocity field, its spatial gradient, and the pressure gradient exhibit simultaneous localized amplification at each prescribed point zj. More precisely, each component of the velocity field is bounded below by M, each component of the pressure gradient is of size at least M2γ/(γ−1), and the velocity gradient exhibits a prescribed anisotropic sign pattern with entries of size at least M at every zj. All implicit constants in these estimates are independent of M and depend solely on the fixed physical parameters and the locations {zj}j=1N. This mechanism generates highly localized, shock-like gradient concentrations and implosion-like spatial profiles within the smooth regime. The underlying mechanism relies on the construction of the Herglotz kernel via Helmholtz far-field patterns, together with delicate quantitative energy estimates for the quasilinear hyperbolic system.
Related
- Existence and stability of shock profiles for a one-dimensional full compressible Navier--Stokes--Poisson equations
- Structurally stable singularities and Lipschitz stable optimal transport metrics for the compressible Euler equations
- Uniqueness and stability of planar shock wave to 2D compressible Euler equations in the vanishing viscosity limit of compressible Navier-Stokes equations
- Isolated singularities of certain quasi-linear elliptic inequalities II
- Development of singular surfaces and formation of generic singularities in 3D pressureless flows
BibTeX
@misc{fan2026formation,
title = {{Formation of quasi-singularities of shock and implosion type for compressible Euler flows}},
author = {Xieling Fan and Hongyu Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.29454},
eprint = {2606.29454},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.29454},
}
RIS
TY - UNPB
TI - Formation of quasi-singularities of shock and implosion type for compressible Euler flows
AU - Xieling Fan
AU - Hongyu Liu
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2606.29454
UR - https://arxiv.org/abs/2606.29454
AB - This paper investigates a novel quasi-singularity formation phenomenon in the isentropic compressible Euler equations in three dimensions. For any prescribed finite set of points {z_j}_j=1^N ⊂ ℝ^3 and an arbitrarily large parameter M ≫ 1, we explicitly construct real-analytic initial data generated by Herglotz-type wave fields. We prove that the corresponding classical solution (p,v)∈ C^1([0,T]×ℝ^3;ℝ×ℝ^3), with T = C_0 M^−(3γ+1)/(γ−1), where γ> 1 is the adiabatic index and C_0 > 0 is a constant independent of M. Throughout the time interval $[0, T]$, the velocity field, its spatial gradient, and the pressure gradient exhibit simultaneous localized amplification at each prescribed point z_j. More precisely, each component of the velocity field is bounded below by M, each component of the pressure gradient is of size at least M^2γ/(γ−1), and the velocity gradient exhibits a prescribed anisotropic sign pattern with entries of size at least M at every z_j. All implicit constants in these estimates are independent of M and depend solely on the fixed physical parameters and the locations {z_j}_j=1^N. This mechanism generates highly localized, shock-like gradient concentrations and implosion-like spatial profiles within the smooth regime. The underlying mechanism relies on the construction of the Herglotz kernel via Helmholtz far-field patterns, together with delicate quantitative energy estimates for the quasilinear hyperbolic system.
ER -