Research of the Riccati equation for the one-dimensional Godunov-Sultangazin system

Abstract

The study of kinetic nonlinear hyperbolic partial differential equations at large times belongs to the field of mathematical physics that has been actively developing recently.

The kinetic theory considers gas as a combination of a huge number of moving particles interacting with each other. As a result of such interactions, the particles exchange momentum and energy. The interaction can be carried out by direct collision or by other forces. To describe the above assumptions, a number of models are proposed — the so-called discrete kinetic equations of Carleman, Godunov-Sultangazin, Broadwell where the unknown functions are particles densities depending on the space-time coordinates.

In this article the Riccati equation for the zero mode is researched obtained from the system of kinetic equations of Godunov-Sultangazin with periodic initial data. The system describes three groups of particles moving at three speeds. The first group moves at unit speed in a positive direction, and the third in the opposite direction. Particles of the second group move at zero speed. The solution of the system is found near the equilibrium state with small periodic perturbations. The solution of the Riccati equation is sought by the method of successive approximations. These perturbations are Fourier series. Theorems of global existence and uniqueness of the solution of the Riccati equation are proved.