Existence of a global weak solution for a 2d viscous bi-layer Shallow Water model

We consider a non-linear viscous bi-layer shallow water model with capillarity effects and extra friction terms in a two-dimensional space. This system is issued from a derivation of three-dimensional Navier–Stokes equations with a water-depth depending on friction coefficients. We prove an existence result for a global weak solution in a periodic domain Ω = T 2