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

Abstract

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 $({\partial }_x{u}_0){\partial }_x{v}_0$ belongs to $L^2(\mathbb{R})$. We construct a global semigroup of conservative weak solutions. We also discuss the potential concentration phenomena of $({\partial }_{x}u{)}^{2}dx$, $({\partial }_{x}v{)}^{2}dx$, and $\left(({\partial }_{x}u{)}^2({\partial }_{x}v{)}^2\right)dx$, 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.