This note is devoted to the study of a bi-fluid generalization of the nonlinear shallow-water equations. It describes the evolution of the interface between two fluids of different densities. In the case of a two-dimensional interface, this systems contains unexpected nonlocal terms (that are of course not present in the usual one-fluid shallow water equations). We show here how to derive this systems from the two-fluid Euler equations and then show that it is locally well-posed.

Building upon the techniques introduced in [15], for any $\theta <\frac{1}{10}$ we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are Hölder-continuous with exponent $\theta$. A famous conjecture of Onsager states the existence of such dissipative solutions with any Hölder exponent $\theta <\frac{1}{3}$. Our theorem is the first result in this direction.