Blow-up for the two-component Camassa-Holm system

Following conservative solutions of the two-component Camassa-Holm system $u_t-u_{txx}+3uu_x-2u_xu_{xx}-uu_{xxx}+ρρ_x=0$, $ρ_t+(uρ)_x=0$ along characteristics, we determine if wave breaking occurs in the nearby future or not, for initial data $u_0\in H^1(\mathbb R)$ and $ρ_0\in L^2(\mathbb R)$.

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