A Hyperbolic Model of Granular Flow
Debora Amadori and Wen Shen
Abstract. In this paper we review some recent results for a model for granular
flow that was proposed by Hadeler & Kuttler in [20]. In one space dimension,
this model can be written as a 2 × 2 hyperbolic system of balance laws, in
which the unknowns represent the thickness of the moving layer and the one
of the resting layer.
If the slope does not change its sign, the resulting system can be analyzed
by means of the known theory, as for instance in the context of small
C1
data
or small BV data. Moreover, due to the special hyperbolicity properties of the
system and of the special form of the source term, it is possible to enlarge the
class of initial data for which global in time solutions exist. See [2, 29].
Further, we study the “slow erosion/deposition limit”, [3], where the
thickness of the moving layer vanishes but the total mass of flowing down
material remains positive. The limiting behavior for the slope of the mountain
profile provides an entropy solution to a scalar integro-differential conservation
law.
A well-posedness analysis of this integro-differential equation is presented.
Therefore, the solution found in the limit turns out to be unique.
Contents
1. The model of granular flow 2
2. Global smooth solutions 4
3. Global existence of large BV solutions 5
4. Global BV solutions of an initial boundary value problem 9
5. Slow erosion limit 10
References 17
2000 Mathematics Subject Classification. Primary 35L65, 35L50, 76T25.
The authors wish to thank Professors H. Holden and K.H. Karlsen for their kind invitation
to the international research program on Nonlinear Partial Differential Equations, organized at
the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during
the academic year 2008–09.
1
Contemporary Mathematics
Volume 526, 2010
c 2010 American Mathematical Society
http://dx.doi.org/10.1090/conm/526/10374
Previous Page Next Page