### ON ESTIMATES OF THE LINEARIZED OPERATOR OF THE KINETIC CARLEMAN SYSTEM

Pages 7-14

In this article the author discusses the properties of linearized operator for the Carleman equation for unperturbed problem, i.e. without the perturbation operator. The solution of the Cauchy problem with periodic initial data is searched for small perturbations of the equilibrium state. The estimates are obtained using the Paley-Wiener theorem and the Laplace transformation. It is assumed that the solutions of the Cauchy problem split into the superposition of weakly interacting solutions and decreasing dispersive waves. The Carleman equation describes a combination of processes: relaxation and free movement. The aim of relaxation is to spread the particles in different directions. Such a system simulates some properties of the Boltzmann equation. The kinetic Carleman equation is a system of two nonlinear differential equations describing transportation processes and interaction of two classes of particles moving with the same speed in modulus in different directions on the line. This system belongs to the class of non-integrable equations which leads to important consequences. Namely, such a system can detect the irregular behavior of the solutions. The Carleman system occupies a special position with respect to other systems and allows us to prove the global existence theorem. In the works by Il’in the question of the stability of stationary but spatially inhomogeneous solutions of the Carleman system is posed. In the case of the discrete model the solution is stable in time for the homogeneous problem.

DOI: 10.22227/1997-0935.2016.9.7-14