Some smoothness and uniqueness results for a shallow-water problem

Abstract

In a previous work, we have shown the existence of weak solutions for a shallow-water problem (or compressible two-dimensional Navier-Stokes problem) in a depth-mean velocity formulation. Some results have been proved by Kazhikov in the case of domain equal to $\mathbb{R}^N$ and linearized momentum equation, which allows him to look for a velocity of the form $u=\nabla p$. We present some smoothness and uniqueness results whatever a smooth domain and with the complete momentum equation and boundary conditions on $u \cdot n$ and $\mathcal{u}$.