On the stabilization rate of periodic perturbations of equilibrium states for the one-dimensional Broadwell kinetic equation

The paper deals with a procedure for constructing solutions to the problem of stabilizing periodic perturbations of equilibrium states in the one-dimensional Broadwell model. The solution procedure employs the Fourier method to solve the system of equations for the Fourier coefficients of the variables. In the Fourier transform space, the system reduces to a projection onto a single variable, enabling expression of the remaining Fourier coefficients $u_{k,l}$, $v_{k,l}$, $w_{k,l}$ through $z_{k,l}$ by using state equations.
The linearization of the $z$-projection plays a crucial role in studying the stabilization rate, representing in this case an integro-differential operator described in terms of the Paley–Wiener theorem. The discrepancy between the right and left sides of the one-dimensional system creates obstacles in the Fourier method when constructing annihilators of secular terms for the corresponding projection. These obstacles prevent obtaining solutions for arbitrary initial data describing periodic perturbations of the equilibrium position. It is established that the arising obstacles are identical for different projections.