Blow-up, critical norm inflation, and global well-posedness for the Geng--Xue system
Abstract
We study finite-time blow-up, critical norm inflation, and global well-posedness for the Geng--Xue system, an integrable two-component extension of the Novikov equation. For initial velocities (u0,v0)∈(Hs)2, s>3/2, with second momentum n0=v0−v0,xx∈ L1uloc, we derive explicit blow-up conditions and upper bounds on the lifespan. The blow-up proof uses a cancellation in the equation for nux to close the convolution estimates without an a priori bound on |ux|L∞. Combined with a continuation criterion involving only u, these estimates yield critical Besov norm inflation around the initial data in (H3/2)2. For initial momenta in (Hσ)2, σ>−1/2, each with a fixed sign, we prove global well-posedness. The blow-up and norm-inflation results also extend to the three-component Degasperis--Procesi system.
Related
- Local well-posedness, blow-up phenomenon and ill-posedness for a model system of the radiating gas
- Charge-free well-posedness and charge-driven norm inflation for the Euler-Maxwell system
- Global well-posedness and scattering for mass-critical Hartree equation
- Global well-posedness for the energy-critical focusing NLS on 𝕋3
- Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces
BibTeX
@misc{liu2026blowup,
title = {{Blow-up, critical norm inflation, and global well-posedness for the Geng--Xue system}},
author = {Song Liu and Zhaoyang Yin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34267},
eprint = {2610.34267},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34267},
}
RIS
TY - UNPB
TI - Blow-up, critical norm inflation, and global well-posedness for the Geng--Xue system
AU - Song Liu
AU - Zhaoyang Yin
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34267
UR - https://arxiv.org/abs/2610.34267
AB - We study finite-time blow-up, critical norm inflation, and global well-posedness for the Geng--Xue system, an integrable two-component extension of the Novikov equation. For initial velocities (u_0,v_0)∈(H^s)^2, s>3/2, with second momentum n_0=v_0−v_0,xx∈ L^1_uloc, we derive explicit blow-up conditions and upper bounds on the lifespan. The blow-up proof uses a cancellation in the equation for nu_x to close the convolution estimates without an a priori bound on |u_x|_L^∞. Combined with a continuation criterion involving only u, these estimates yield critical Besov norm inflation around the initial data in (H^3/2)^2. For initial momenta in (H^σ)^2, σ>−1/2, each with a fixed sign, we prove global well-posedness. The blow-up and norm-inflation results also extend to the three-component Degasperis--Procesi system.
ER -