Finite-time blow-up for the Geng--Xue system with an application to the three-component Degasperis--Procesi system

Abstract

We prove finite-time blow-up for the Geng--Xue system with initial data satisfying $u_0\in W^{p,q}(\mathbb{R})$ and $n_0\in L^{r^{\prime }}(\mathbb{R})$ at Sobolev regularity $s\ge 2$, where $n=v-v_{xx}$. A cancellation in the balance law for $n{u}_x$ yields closed convolution estimates that do not depend on an a priori uniform bound for $u_x$. These estimates control $v$ along a characteristic and give an explicit condition on the initial slope, together with an upper bound on the lifespan. The proof does not use conservation laws. For the three-component Degasperis--Procesi system, the same argument applies to $n{u}_x+{\rho }^2-{\bar{\rho }}^2$, where $\bar{\rho }\in [0,1]$ is a constant density background. We also prove local well-posedness for this system in $H^{s-1}(\mathbb{R})\times H^s(\mathbb{R})\times H^s(\mathbb{R})$ for $s>\frac{3}{2}$, and a blow-up criterion in $B_{2,1}^1(\mathbb{R})\times B_{2,1}^2(\mathbb{R})\times B_{2,1}^2(\mathbb{R})$.