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 $(u_0,v_0)\in(H^s)^2$, $s>3/2$, with second momentum $n_0=v_0-v_{0,xx}\in L^1_{\rm uloc}$, 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 $\|u_x\|_{L^\infty}$. 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.