### 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 $\mathcal{u}$.

#### Article information

Source
Adv. Differential Equations, Volume 3, Number 1 (1998), 155-176.

Dates
First available in Project Euclid: 19 April 2013