One-Dimensional Granular System with Memory Effects

Abstract

We consider a hybrid compressible/incompressible system with memory effects, introduced recently by Lefebvre Lepot and Maury for the description of one-dimensional granular flows. We prove a global existence result for this system without assuming additional viscous dissipation. Our approach extends the one by Cavalletti et al. for the pressureless Euler system to the constrained granular case with memory effects. We construct Lagrangian solutions based on an explicit formula using the monotone rearrangement associated to the density. We explain how the memory effects are linked to the external constraints imposed on the flow. This result can also be extended to a heterogeneous maximal density constraint depending on time and space.

Finite volume approximations of the Euler system with variable congestion

“…This proposition can be proven using similar arguments as in [16]. Below, on Figure 11 we present two different solutions to the Riemann problem (29)- (30). Depending on the initial location of the left and right states, the intersection state (v m , Z m ) might be a congested state or not.…”

“…The purpose of this section is to find possible weak solution to (29) (30). We will also consider the limit of these solutions as ε → 0.…”

“…Let ( , q , Z ) and ( r , q r , Z r ) be the left and right initial states (30). The solutions to Riemann problems are determined as follows.…”

“…introduced originally by Bouchut et al [8], who also proposed the first numerical scheme based on an approach developed earlier for the pressureless systems, see for example [9], and the projection argument. The model was studied later on by Berthelin [3,4] by passing to the limit in the so-called sticky-blocks dynamics, see also [35], and a very interesting recent paper [30] using the Lagrangian approach for the monotone rearrangement of the solution to prove the existence of solutions to (6) with additional memory effects. The pressureless Euler equations with the density constraint were originally introduced in order to describe the motion of particles of finite size.…”

A remark on memory effects in constrained fluid systems

“…memory effects, that may be activated in the congested domain (see the system (3) presented in the next section). These memory effects, encoded in the so-called adhesion potential, were introduced formally for the free-congested Euler system by Lefebvre-Lepot and Maury [16] and justified from the mathematical viewpoint in the recent studies [21], [24], [4]. These results are reviewed in the previous paper [23].…”