We consider in this talk the Serre-Green-Naghdi equations with surface tension. Those equations are locally (in time) well-posed, we present a precise blow-up criterion and we identify a class of smooth initial data such that the corresponding strong solutions develop singularities in finite time. We also show the existence of a strongly continuous semigroup of global weak dissipative solutions for any small energy initial data. The Riemann invariants of the weak solutions satisfy a one-sided Oleinik inequality.