Global Existence of Large BV Solutions in a Model of Granular Flow

Abstract

In this paper we analyze a set of equations proposed by Hadeler and Kuttler [20], describing the flow of granular matter in terms of the heights of a standing layer and of a moving layer. By a suitable change of variables, the system can be written as a 2 × 2 hyperbolic system of balance laws, which we study in the one-dimensional case. The system is linearly degenerate along two straight lines in the phase plane, and therefore is weakly linearly degenerate at the point of the intersection. The source term is quadratic, consisting of product of two quantities, which are transported with strictly different speeds. Assuming that the initial height of the moving layer is sufficiently small, we prove the global existence of entropy-weak solutions to the Cauchy problem, for a class of initial data with bounded but possibly large total variation.

Erosion Profile by a Global Model for Granular Flow

“…If x i is a concave kink with w i+ 1 2 < w i− 1 2 , then it is small. When k ≤ u i and l > ζ(w i+ 1 2 ), the error introduced is bounded by a quadratic term of the wave strength. We have Here in the last inequality we used the estimate (5.29).…”

“…If x i is at the right side of a shock, then, the Lax condition is only approximated. For u i−1 < k < u i and l > ζ(w i+ 1 2 ), we have (P ε,+ − P ε,− )ẋ i − (Q ε,+ F i − Q ε,− F i−1 )…”

“…Since convex kinks are admissible, we do not have any restriction on the slopes w i+ 1 2 , w i− 1 2 . By (5.12) we havė…”

“…This happens where w crosses 1 in a rarefaction fan. By construction, since the slope w = 1 is always selected for some interval in such a rarefaction fan, we must have another neighbor point, say U i+1 , such that U i = U i+1 , so w i+ 1 2 = 1 and f (w i+ 1 2 ) = 0. Then, by (5.12) we haveU i =U i+1 = 0, and the local maximum does not change its value.…”

“…In [24] an existence result for global smooth solutions is proved. The paper [1] establishes the global existence to the Cauchy problem with large data, within a class of functions with bounded variation. Furthermore, in [2] the authors prove the global existence of large BV solutions for an initial-boundary value problem.…”