On discrete models of Boltzmann-type kinetic equations

The known nonlinear kinetic equations, in particular, the wave kinetic equation and the quantum Nordheim–Uehling–Uhlenbeck equations are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables $F(x,y; v,w)$. The function $ F $ is assumed to satisfy certain simple relations. The main properties of this kinetic equation are studied. It is shown that the above mentioned specific kinetic equations correspond to different polynomial forms of the function $ F $. Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas similar to those used for construction of discrete velocity models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similarly to the Boltzmann $ H $-function. The theorem of existence, uniqueness and convergence to equilibrium of solutions to the Cauchy problem with any positive initial conditions is formulated and discussed. The differences in long time behaviour between solutions of the wave kinetic equation and solutions of its discrete models are also briefly discussed.