On a two-velocity model of the Boltzmann-Enskog equation (Q1098714)

A two-velocity model of the Boltzmann-Enskog equation for numbers densities \(N_ 1(t,x)\), \(N_ 2(t,x)\) has the form: \[ D_+N_ 1(t,x)=N_ 1(t,x-\sigma)N_ 2(t,x)-N_ 1(t,x)N_ 2(t,x+\sigma) \] \[ D_-N_ 2(t,x)=N_ 1(t,x)N_ 2(t,x+\sigma)-N_ 1(t,x-\sigma)N_ 2(t,x) \] supplemented by nonnegative initial data \(N_ i(0,x)=N_{i0}(x)\), \(i=1,2\), \(x\in R\), \(t\in R_ t\); \(D_{\pm}=\partial_ t\pm \partial_ x\). This model describes a fictitious gas of hard spheres with diameter \(\sigma >0\) having discrete \(+1\) and -1 velocities. 3 global existence theorems and some qualitative properties of solutions of the model are proved.