Multi-dimensional centered rarefaction waves of gas dynamics with vacuum states
Abstract
We establish the local-in-time structural stability of two centered rarefaction waves bounding a vacuum region for the two-dimensional isentropic Euler equations with p(ρ)=kργ, for γ∈(1,1.3]∪{7/5,5/3}. For general sufficiently small perturbations of the planar Riemann data, allowing nonzero vorticity, we construct an entropy solution that is continuous for positive time and whose five regions are separated by four characteristic surfaces. The conservative solution has uniform spatial W2,∞ bounds on each time slice of the central region in coordinates (x1/t,x2); for γ=7/5,5/3, we also obtain spacetime W2,∞ regularity in (t,x1/t,x2). The proof develops characteristic geometry directly for the symmetric first-order Euler system and uses two overlapping characteristic charts to separate matching with the exterior flow from estimates uniform up to vacuum. Fixed commutators without sound-speed weights exhibit a first-order null structure, while subtraction of explicit approximate solutions produces damping that cancels the leading antidamping bulk terms in the energy identity. An explicit analysis of the initial-slice ODEs identifies fractional powers and logarithmic terms at zero sound speed. The logarithmic coefficients couple vorticity to tangential variation of the initial data, revealing an obstruction to higher spatial regularity even when the powers of the sound speed have integer exponents. This analysis determines the finite approximation order used to close the energy and geometric estimates in the stated range. These ingredients provide the geometric and energy framework for the forthcoming second paper of this series, which will use finite evolving profiles to treat the full range γ∈(1,5/3].
Related
- Time-Decay Rates toward Rarefaction Waves for the One-Dimensional Isentropic Compressible Navier--Stokes Equations
- Vacuum Structure of Maximally Supersymmetric Two-Dimensional Gauge Theories
- Global Well-Posedness of Strong Solutions to the Two-Dimensional Compressible Nematic Liquid Crystal Flows with Large Initial Data and Vacuum
- Global Existence and Large-Time Behavior of Strong Solutions to the Two-Dimensional Compressible Nematic Liquid Crystal Flows with Large Initial Data and Vacuum
- Global Strong Solutions to the One-Dimensional Planar Compressible Magneto-Micropolar Equations with Vacuum
BibTeX
@misc{jia2026multidimensional,
title = {{Multi-dimensional centered rarefaction waves of gas dynamics with vacuum states}},
author = {Jin Jia and Tao Luo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18177},
eprint = {2610.18177},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18177},
}
RIS
TY - UNPB
TI - Multi-dimensional centered rarefaction waves of gas dynamics with vacuum states
AU - Jin Jia
AU - Tao Luo
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.18177
UR - https://arxiv.org/abs/2610.18177
AB - We establish the local-in-time structural stability of two centered rarefaction waves bounding a vacuum region for the two-dimensional isentropic Euler equations with p(ρ)=kρ^γ, for γ∈(1,1.3]∪{7/5,5/3}. For general sufficiently small perturbations of the planar Riemann data, allowing nonzero vorticity, we construct an entropy solution that is continuous for positive time and whose five regions are separated by four characteristic surfaces. The conservative solution has uniform spatial W^2,∞ bounds on each time slice of the central region in coordinates (x_1/t,x_2); for γ=7/5,5/3, we also obtain spacetime W^2,∞ regularity in (t,x_1/t,x_2). The proof develops characteristic geometry directly for the symmetric first-order Euler system and uses two overlapping characteristic charts to separate matching with the exterior flow from estimates uniform up to vacuum. Fixed commutators without sound-speed weights exhibit a first-order null structure, while subtraction of explicit approximate solutions produces damping that cancels the leading antidamping bulk terms in the energy identity. An explicit analysis of the initial-slice ODEs identifies fractional powers and logarithmic terms at zero sound speed. The logarithmic coefficients couple vorticity to tangential variation of the initial data, revealing an obstruction to higher spatial regularity even when the powers of the sound speed have integer exponents. This analysis determines the finite approximation order used to close the energy and geometric estimates in the stated range. These ingredients provide the geometric and energy framework for the forthcoming second paper of this series, which will use finite evolving profiles to treat the full range γ∈(1,5/3].
ER -