Ill-posedness of Dynamic Equations of Compressible Granular Flow

We introduce models for 2-dimensional time-dependent compressible flow of granular materials and suspensions, based on the \begin{figure}[htbp] \centerline{\includegraphics[width=0.22in,height=0.17in]{280720171.eps}} \label{fig1} \end{figure} rheology of Pouliquen and Forterre. The models include density dependence through a constitutive equation in which the density \begin{figure}[htbp] \centerline{\includegraphics[width=0.13in,height=0.17in]{280720172.eps}} \label{fig2} \end{figure} or volume fraction of solid particles with material density $\rho $* is taken as a function of an inertial number I: $\rho =\rho $*$\Phi $(I), in which $\Phi $(I) is a decreasing function of I. This modelling has different implications from models relying on critical state soil mechanics, in which $\rho $ is treated as a variable in the equations, contributing to a flow rule. The analysis of the system of equations builds on recent work of Barker et al in the incompressible case. The main result is the identification of a criterion for well-posedness of the equations. We additionally analyze a modification that applies to suspensions, for which the \begin{figure}[htbp] \centerline{\includegraphics[width=0.22in,height=0.17in]{280720173.eps}} \label{fig3} \end{figure} rheology takes a different form and the inertial number reflects the role of the fluid viscosity.