Global weak solutions of the Serre-Green-Naghdi equations with surface tension

We consider in this paper the Serre–Green–Naghdi equations with surface tension. Smooth solutions of this system conserve an $H^1$-equivalent energy. We prove the existence of global weak dissipative solutions for any relatively small-energy initial data. We also prove that the Riemann invariants of the solutions satisfy a one-sided Oleinik inequality.