On a speed of solutions stabilization of the Cauchy problem for the Carleman equation with periodic initial data

This article explores a one-dimensional system of equations for the discrete model of a gas (Carleman system of equations). The Carleman system is the Boltzmann kinetic equation of a model one-dimensional gas consisting of two particles. For this model, momentum and energy are not retained. On the example of the Carleman model, the essence of the Boltzmann equation can be clearly seen. It describes a mixture of “competing” processes: relaxation and free movement. We prove the existence of a global solution of the Cauchy problem for the perturbation of the equilibrium state with periodic initial data. For the first time we calculate the stabilization speed to the equilibrium state (exponential stabilization).