Geodesic
On problem of nonexistence of dissipative estimate for discrete kinetic equations
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 130 (2013) no. 1, 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. 
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, Tome 130 (2013) no. 1, pp. 106-143. http://geodesic.mathdoc.fr/item/VSGTU_2013_130_1_a11/
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