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 $H^1\left(R\right)$ such that the product $\left({\partial }_x{u}_0\right){\partial }_x{v}_0$ belongs to $L^2\left(R\right)$. We construct a global semigroup of conservative weak solutions. We also discuss the potential concentration phenomena of ${\left({\partial }_{x}u\right)}^{2}dx$, ${\left({\partial }_{x}v\right)}^{2}dx$, and $\left({\left({\partial }_{x}u\right)}^2{\left({\partial }_{x}v\right)}^2\right)dx$, 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.