Geodesic
On problem of nonexistence of dissipative estimate for discrete kinetic equations
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2013), pp. 106-143.

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The 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.
Keywords: dissipative estimates, discrete kinetic equations. 
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E. V. Radkevich. On problem of nonexistence of dissipative estimate for discrete kinetic equations. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2013), pp. 106-143. http://geodesic.mathdoc.fr/item/VSGTU_2013_1_a11/

[1] E. V. Radkevich, “The existence of global solutions to the cauchy problem for discrete kinetic equations”, J. Math. Sci., New York, 181:2 (2012), 232-280 | DOI | MR | Zbl

[2] T. E. Broadwell, “Study of rarified shear flow by the discrete velocity method”, J. Fluid Mech., 19:3 (1964), 401–414 | DOI | MR | Zbl

[3] S. K. Godunov, U. M. Sultangazin, “On discrete models of the kinetic Boltzmann equation”, Russian Math. Surveys, 26:3 (1971), 1–56 | DOI | MR | Zbl

[4] L. Boltzmann, “On the Maxwell method to the reduction of hydrodynamic equations from the kinetic gas theory”, Rep. Brit. Assoc. , London, 1894, 579; L. Boltsman, “O maksvellovskom metode vyvoda uravnenii gidrodinamiki iz kineticheskoi teorii gazov”, Izbrannye trudy, Molekulyarno-kineticheskaya teoriya gazov. Termodinamika. Statisticheskaya mekhanika. Teoriya izlucheniya. Obschie voprosy fiziki, ed. L. S. Polak, Nauka, M., 1984, 307–308 | MR

[5] V. V. Vedenyapin, Kinetic Boltzmann and Vlasov Equations, Fizmatlit, Moscow, 2001, 112 pp.

[6] 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. | MR

[7] R. Peierls, “Zur kinetischen Theorie der Wärmeleitung in Kristallen”, Ann. Phys., 395:8 (1929), 1055–1101 | DOI | Zbl