Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical Sciences1991-86152310-7081Samara State Technical University3469910.14498/vsgtu1140Research ArticleOn problem of nonexistence of dissipative estimate for discrete kinetic equationsRadkevichEvgenii VladimirovichDoctor of physico-mathematical sciences, Professorevrad07@gmail.comLomonosov Moscow State University, Faculty of Mechanics and Mathematics1512201317110614310062020Copyright © 2013, Samara State Technical University2013The existence of a global solution to the discrete kinetic equations in Sobolev spaces is proved, its decomposition by summability is obtained, the influence of its oscillations generated by the interaction operator is explored. The existence of a submanifold ${\mathcal M}_{diss}$ of initial data $(u^0, v^0, w^0)$ for which the dissipative solution exists is proved. It’s shown that the interaction operator generates the solitons (progressive waves) as the nondissipative part of the solution when the initial data $(u^0, v^0, w^0)$ deviate from the submanifold ${\mathcal M}_{diss}$. The amplitude of solitons is proportional to the distance from $(u^0, v^0, w^0)$ to the submanifold ${\mathcal M}_{diss}$. It follows that the solution can stabilize as $t\to\infty$ only on compact sets of spatial variables.dissipative estimatesdiscrete kinetic equationsдиссипативные оценкидискретные кинетические уравнения[Е. В. Радкевич, "О существовании глобальных решений задачи Коши для дискретных кинетических уравнений (непериодический случай)", Пробл. мат. анал., 62 (2012)][T. E. Broadwell, "Study of rarified shear flow by the discrete velocity method", J. Fluid Mech., 19:3 (1964), 401-414][С. К. Годунов, У. М. Султангазин, "О дискретных моделях кинетического уравнения Больцмана", УМН, 26:3(159) (1971), 3-51][L. Boltzmann, "On the Maxwell method to the reduction of hydrodynamic equations from the kinetic gas theory", Rep. Brit. Assoc. , London, 1894, 579][В. В. Веденяпин, Кинетические уравнения Больцмана и Власова, Физматлит, М., 2001, 112 с.][S. Chapman, T. G. Cowling, The mathematical theory of non-uniform gases. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, Cambridge University Press, Cambridge, 1970, xxiv+423 pp.][R. Peierls, "Zur kinetischen Theorie der Wärmeleitung in Kristallen", Ann. Phys., 395:8 (1929), 1055–1101]