Secular Terms for the Kinetic Mckean Model

Abstract

In this article, we investigate the kinetic McKean model. The perturbed solution of the Cauchy problem is sought in the form of Fourier series. The Fourier coefficients for the zero and nonzero modes are written out, respectively. The original system is reduced to an infinite system of differential equations. An approximation for the systems is constructed. Under certain assumptions, we find secular terms (non-integrable part). This, in turn, will allow us to prove for the first time the exponential stabilization of the solution in the future.

A self-similar solution for the two-dimensional Broadwell system via the Bateman equation

“…The physical interpretation of the system can be found in [7,20]. Works [1,5,11] are devoted to finding exact solutions of kinetic systems by means of the Bateman equation [3,6,13,14]. These systems are non-integrable (kinetic systems Carleman [1], Godunov-Sultangazin [9,11] (onedimensional model of Broadwell), McKean [5,12], two-dimensional model of Broadwell).…”