4 D. AMADORI AND W. SHEN
is capable to describe approximately a complex dynamics like the evolution of two
layers granular matter. A mathematical analysis of steady state solutions for (1.1)
was carried out in [9, 10]; a numerical study has been performed in .
Other models can be found in [6, 16, 27]. We refer also to the recent paper
 where an extension of the present model is proposed; there, it is introduced
an additional equation for the velocity of the sliding material, leading to a reﬁned
description of the complete dynamics.
Note that, besides , the papers [2, 3] provide the ﬁrst analytical study of
time dependent solutions to this system.
2. Global smooth solutions
The global existence of smooth solutions is established in , under suitable
assumptions on the initial data.
By a decoupled initial data we mean a set of initial conditions of the form
(2.1) h(0,x) = φ(x) p(0,x) = 1 + ψ(x)
with φ, ψ satisfying
φ(x) = 0 if x / ∈ [a, b] ,
ψ(x) = 0 if x / ∈ [c, d] .
The intervals are disjoint, i.e., a b c d. Moreover we assume ψ(x) −1 for
For decoupled initial data, a global solution of the Cauchy problem can be
explicitly given, namely
h(t, x) = φ(x + t) , p(t, x) = 1 + ψ(x) , x ∈ I R , t ≥ 0 .
Our ﬁrst result provides the stability of these decoupled solutions. More pre-
cisely, every suﬃciently small, compactly supported perturbation of a Lipschitz
continuous decoupled solution eventually becomes decoupled. Moreover, no gradi-
ent catastrophe occurs, i.e., solutions remain smooth for all time.
Theorem 2.1 (Global smooth solutions). Let a b c d be given, together
with Lipschitz continuous, decoupled initial data as in (2.1). Then there exists δ 0
such that the following holds. For every perturbations
ψ = 0 if x / ∈ [a, d] ,
φ (x)| ≤ δ , |
ψ (x)| ≤ δ ,
the Cauchy problem for (1.2) with initial data
(2.3) h(0,x) = φ(x) +
φ , p(0,x) = 1 + ψ(x) +
has a unique solution, deﬁned for all t ≥ 0 and globally Lipschitz continuous. More-
over, this solution becomes decoupled in ﬁnite time.
The proof relies on the method of characteristics . One must bound the
norms of hx and px. Since we are looking for continuous solutions, it
is convenient to work in a set of Riemann coordinates. Let (h, p) → (w, z) be the
coordinate transformation such that
(w, z)(h, 1) = (h, 0) for all h , (w, z)(0,p) = (0, p − 1) for all p 0 ,
and r1 • z ≡ 0 , r2 • w = 0. In these new variables, the system (1.2) takes the form
wt + λ1(w, z) wx = f(w, z) ,
zt + λ2(w, z)zx = g(w, z) ,