Global semigroup of conservative weak solutions of the two-component Novikov equation

We study the Cauchy problem for the two-component Novikov system with initial data $u_0,v_0$ in $H^1(\mathbb{R})$ such that the product $({\mathrm{\partial }}_x{u}_0){\mathrm{\partial }}_x{v}_0$ belongs to $L^2(\mathbb{R})$. We construct a global semigroup of conservative weak solutions. We also discuss the potential concentration phenomena of $({\mathrm{\partial }}_{x}u{)}^{2}dx$, $({\mathrm{\partial }}_{x}v{)}^{2}dx$, and $(({\mathrm{\partial }}_{x}u{)}^2({\mathrm{\partial }}_{x}v{)}^2)dx$, which contribute to wave-breaking and may occur for a set of time with nonzero measure. Finally, we establish the continuity of the data-to-solution map in the uniform norm.