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The Vlasov kinetic equation, dynamics of continuum and turbulence. (English) Zbl 1257.37016
Based on the reversible Poisson-Vlasov type kinetic equation, the author proposes a new approach to deriving hydrodynamic type equations for a continuum medium of interacting particles. The main point underlined by the author is related to the irreversibility of the latter versus to the reversibility of the basic Poisson-Vlasov type equation. As the Bogolubov correlation weakening principle cannot be rigorously proved, owing to M. Kac, the author argues that his approach solves this irreversibility problem for the macroscopic behavior of the continuum medium under regard. This solution is based on the analysis of the properties of weak limits as \( t\rightarrow \infty \) of solutions to the Poisson-Vlasov-type equation: \[ \partial \rho ^{\ast }/\partial t+u_{i}\partial \rho ^{\ast }/\partial x_{i}-\partial \rho ^{\ast }/\partial u_{j}\frac{\partial }{\partial x_{j}} \int \int K(x,y)\rho ^{\ast }(y,w,t)dwdy=0 \tag{1} \] for the density \(\rho ^{\ast }\) of the continuum medium, which possesses the exact generalized solution \[ \rho ^{\ast }(x,u,t)=\sum_{i=1}^{n}m_{i}\delta (x-x_{i})\delta (u-u_{i}), \tag{2} \] where \(m_{i}\) are particles masses. The equation (1) is a specified form of the original Vlasov equation \[ \frac{\partial f}{\partial t}+\frac{\partial f}{\partial x_{i}}u_{i}+\frac{ dp_{i}}{dt}\frac{\partial f}{\partial p_{j}}=0, \tag{3} \] see [A. A. Vlasov, Many-particle theory and its application to plasma. New York: Gordon and Breach Science Publishers (1961); “The vibrational properties of an electron gas”, Sov. Phys. Usp. 10, 721–733 (1968)].
In Section 2, Euler equations of motion are derived by means of averaging solutions to the Poisson-Vlasov type equation (1). It is mentioned that the obtained equations are not closed depending on the stress tensor. Section 3 is devoted to deriving an infinite chain of Benney type moment equations. In Sections 4–6, the author analyzes the conservation laws and their dissipative properties, based on evolution of the moment of inertia functional. Section 7 is devoted to stationary solutions of the Vlasov type equation (1) making use of the classical Hilbert method. It is found that a stationary solution possesses an arbitrary function \(f(\frac{1}{2} \sum_{i=1}^{n}u_{i}^{2})\), in contrast to the classical Maxwell-Gibbs-type solution, usually used in applications.
In Section 8, the author argues that the approach to studying turbulent motion of a continuous medium, based on the kinetic equations, is more suitable than the one based on Navier-Stokes equations, as it was mentioned earlier by M. Kac. Some considerations of the turbulent motions models, based on the Poisson-Vlasov equation, are analyzed in detail for different interaction potentials between particles.
A short remark concerning the exact solution (2) to the Vlasov type equation (1): the Vlasov kinetic equation (3) was a priori derived by Vlasov [1961, loc. cit.] for a system of identical particles, conserving the particles replacement symmetry property \( (x_{i}\leftrightarrows x_{j}),\) for the distribution function \(\rho ^{\ast }, \) giving rise to the condition that all particles masses should be equal, that is \(m_{i}=m_{j},i,j=\overline{1,n}.\)

MSC:
37A60 Dynamical aspects of statistical mechanics
35Q83 Vlasov equations
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
82B30 Statistical thermodynamics
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