Global weak solutions for a periodic two-component Hunter-Saxton system (Q2892170)

The paper is dealing with the Hunter-Saxton (HS) hydrodynamic system, NEWLINE\[NEWLINE\begin{aligned} u_{txx} + 2u_xu_{xx} + uu_{xxx} - \rho\rho_x = 0,\\ \rho_t + (\rho u)_x = 0.\end{aligned} NEWLINE\]NEWLINE In the case of \(\rho=0\), the respective Hunter-Saxton (HS) equation is a short-wave asymptotic form of the Camassa-Holm (CH) equation for large-amplitude water waves. Both the CH and HS equations are integrable. On the contrary to the classical integrable Korteweg-de Vries equation for small-amplitude waves, the CH and SH equations describe the wave-breaking phenomenon, and, accordingly, the generic solution may lead to a blowup in a finite time. The present paper produces a rigorous proof of the global existence of solutions to the HS system with smooth initial data. To this end, the existence of strong solutions is proven first, and then the existence of weak solutions is proven as the limit case of the theorem for the strong ones.