2 D. AMADORI AND W. SHEN
1. The model of granular flow
The following model was proposed in  to describe granular flows
ht = div(h∇u) − (1 − |∇u|)h ,
1 − |∇u|)h .
These equations describe conservation of masses. The material is divided in two
parts: a moving layer with height h on top and a standing layer with height u at the
bottom. The moving layer slides downhill, in the direction of steepest descent, with
speed proportional to the slope of the standing layer. If the slope |∇u| 1 then
grains initially at rest are hit by rolling grains of the moving layer and start moving
as well. Hence the moving layer gets bigger. On the other hand, if |∇u| 1, grains
which are rolling can be deposited on the bed. Hence the moving layer becomes
Figure 1. A mass of “snow” flowing down. The thick/thin line
corresponds to the proﬁle before/after the “snow” has flown.
This model is studied in one space dimension in the rest of the paper. Deﬁne
= ux, and assume p ≥ 0, one can rewrite (1.1) into the following 2 × 2 system of
− (hp)x = (p − 1)h ,
(p − 1)h
= 0 .
Writing the system of balance laws (1.2) in quasilinear form, the corresponding
Jacobian matrix is computed as
A(h, p) =
p − 1 h
For h ≥ 0 and p 0, one ﬁnds two real distinct eigenvalues λ1 0 ≤ λ2, as
h − p ± (p − h)2 + 4h .
When h is small, i.e., with h ≈ 0, we have
λ1 = −p +
p − 1
, λ2 =