A note on the well-posedness of the one-dimensional multilayer shallow water model

In this short note, we prove by elementary means that the one-dimensional multilayer shallow-water (or Saint-Venant) model for density-stratified fluids is well-posed, provided that (i) the density stratification is stable (i.e. the denser fluid is deeper); (ii) each layer has non-vanishing depth; (iii) the shear velocities are small enough.