We introduce a new model for shallow water flows with non-flat
bottom. A prototype is the Saint Venant equation for rivers and coastal
areas, which is valid for small slopes. An improved model, due to
Savage-Hutter, is valid for small slope variations. We introduce a new
model which relaxes all restrictions on the topography. Moreover it
satisfies the properties (i) to provide an energy dissipation
inequality, (ii) to be an exact hydrostatic solution of Euler
equations.
The difficulty we overcome here is the normal dependence of the
velocity field, that we are able to establish exactly. Applications we
have in mind concern, in particular, computational aspects of flows of
granular material (for example in debris avalanches) where such models
are especially relevant.

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