Existence and stability analysis for the Carleman kinetic system

Kinetic model of the spatio-temporal turbulence

“…One can see that these values can be uniquely determined through the boundary conditions. Earlier in [7] it has been proved that the steady solution can be unstable. We will consider non-positive solutions of the Carleman system because they can be unstable.…”

“…Also the investigation of the Carleman equation is very attractive because space non-uniform stationary solutions can be obtained analytically. It has been proved in [7] that stationary solutions of the boundary problem for the Carleman equation can be linear unstable for a certain range of the outer parameters. For the discovering the chaotic regimes we take into account the nonlinear effects by the use of the numerical solutions.…”

“…The steady solution of the mentioned problem has the following form (see [7]): The steady solution depends on two constants A and C . One can see that these values can be uniquely determined through the boundary conditions.…”

Spatio-temporal Unstable Chaotic Solutions of the Carleman Kinetic System

“…Also the investigation of the Carleman equation is very attractive because space non-uniform stationary solutions can be obtained analytically. It has been proved in [7] that stationary solutions of the boundary problem for the Carleman equation can be linear unstable for a certain range of the outer parameters. The Carleman system (see [8]) in the ordinary dimensionless form is as follows…”

“…Note that it can be easily proved that the global equilibrium solution of the Carleman system, namely, for u=v is stable at least for u>0, v>0. The steady solution of the mentioned problem has the following form (see [7]):…”