A HYPERBOLIC MODEL OF GRANULAR FLOW 3
Note that if p 0, then the two families travel with strictly different speed. A
direct computation gives
r1 λ1 =
2(λ1
+ 1)
λ2 λ1

2(p 1)
p
, r2 λ2 =
2λ2
λ2 λ1
−2
h
p2
,
where r1,r2 are the corresponding eigenvectors, and the “•” stands for the direc-
tional derivative.
This shows the fact that the first characteristic field is genuinely nonlinear away
from the line p = 1 and the second field is genuinely nonlinear away from the line
h = 0, therefore the system is weakly linearly degenerate at the point (h, p) = (0, 1).
Also, the direction of increasing eigenvalues, for the first family, changes with the
sign of p 1. The lines p = 1, h = 0 are characteristic curves of the first, second
family respectively, along which the system becomes linear:
p = 1, ht hx = 0 ; h = 0, pt = 0 .
See Figure 2 for the characteristic curves.
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
2.5
h
p
R
1
R
2
Figure 2. Characteristic curves of the two families in the h-p
plane. The arrows point in the direction of increasing eigenvalues.
In this paper we review some recent results about the existence of solutions
for system (1.2), that are shown to exist globally in time for suitable classes of
initial data. For systems of conservation laws with source term, some dissipation
conditions are known in the literature that ensure the global in time existence of
(smooth or weak) solutions; we refer to [22], to Kawashima–Shizuta condition (see
[21]) for smooth solutions and to [15, 24] for the weak solutions. These conditions
exploit a suitable balance between the differential terms and the source term that
enable to control the nonlinearity of the system. It is interesting to remark that
system (1.2) does not satisfy any of these conditions, nevertheless it admits global
in time solutions.
For a derivation of the model (1.1) of granular flow we refer to [20]. The present
model has recently raised a certain attention in the mathematical community; in-
deed, it exibits interesting properties as a nonlinear model, and at the same time it
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