On a Preserving Loitsyansky Invariant into Millionshtchikov Closure Model of Homogeneous Isotropic Turbulent Dynamics

The existence of a solution to an initial-boundary value problem for Millionshtchikov closure model of the von Kármán–Howarth equation is proven. The behavior of the solution obtained is investigated in the limit of viscosity $\nu$ to zero. We establish the asymptotic stability of the Millionshtchikov selfsimilar solution as $t\to\infty$. Moreover, we prove that Loitsyansky integral plays the role of a conservation law for Millionshtchikov closure model of homogeneous isotropic turbulent dynamics.