# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Bogoliubov, N.N., Problems of Dynamic Theory in Statistical Physics, Oak Ridge, TN: Technical Information Service, 1960. [2] Uhlenbeck, G. E. and Ford, G.W., Lectures in Statistical Mechanics, Providence, RI: AMS, 1963. · Zbl 0111.43802 [3] Chetverushkin, B.N., Kinetic Schemes and Quasi-Gas Dynamic System of Equations, Moscow: Maks Press, 2004 [Barcelona: CIMNE, 2008]. · Zbl 1217.82002 [4] Kac, M., Some Stochastic Problems in Physics and Mathematics, Colloquium lectures in pure and applied science, vol. 2, Magnolia Petroleum Co., 1956. [5] Vlasov, A.A., Statistical Distribution Functions, Moscow: Nauka, 1966 (Russian). [6] Vedenyapin, V.V., Boltzmann and Vlasov Kinetic Equations, Moscow: Fizmatlit, 2001 (Russian). [7] Maslov V.P., Equations of the Self-Consistent Field, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat., vol. 11, Moscow: VINITI, 1978, pp. 153–234] [J. Soviet Math., 1979, vol. 11, pp. 123–195]. [8] Dobrushin, R. L., Vlasov Equations, Funktsional. Anal. i Prilozhen., 1979, vol. 13, no. 2, pp. 48–58 [Funct. Anal. Appl., 1979, vol. 13, no. 2, pp. 115–123]. · Zbl 0422.35068 [9] Arsen’ev, A.A., Some Estimates of the Solution of Vlasov’s Equation, Zh. Vychisl. Mat. Mat. Fiz., 1985, vol. 25, no. 1, pp. 80–87 [USSR Comput. Math. Math. Phys., 1985, vol. 25, no. 1, pp. 52–57]. [10] Kozlov, V.V., The Generalized Vlasov Kinetic Equation // Uspekhi Mat. Nauk, 2008, vol. 63, no. 4(382), pp. 93–130 [Russian Math. Surveys, 2008, vol. 63, no. 4, pp. 691–726]. · Zbl 1181.37006 [11] Benney, D. J., Some Properties of Long Nonlinear Waves, Stud. Appl. Math., 1973, vol. 52, no. 1, pp. 45–50. · Zbl 0259.35011 [12] Zakharov, V.E., Benney Equations and Quasiclassical Approximation in the Method of the Inverse Problem, Funktsional. Anal. i Prilozhen., 1980, vol. 14, no. 2, pp. 15–24 [Funct. Anal. Appl., 1980, vol. 14, no. 2, pp. 89–98]. · Zbl 0455.14022 [13] Gibbons, J., Collisionless Boltzmann Equations and Integrable Moment Equations, Phys. D, 1981, vol. 3, pp. 503–511. · Zbl 1194.35298 [14] Lebedev, D.R. and Manin, Yu. I., Conservation Laws and Lax Representation for Benney’s Long Wave Equation, Phys. Lett. A, 1979, vol. 74, pp. 154–156. [15] Gibbons, J. and Tsarev, S.P., Reductions of the Benney Equations, Phys. Lett. A, 1996, vol. 211, pp. 19–24. · Zbl 1072.35588 [16] Akhiezer, N. I., The Classical Moment Problem and Some Related Questions in Analysis, Edinburgh-London: Oliver & Boyd, 1965. · Zbl 0135.33803 [17] Poincaré, H., Figures d’équilibre d’une masse fluide, Paris: Gauthier-Villars, 1902. · JFM 34.0757.05 [18] Poincaré, H., Réfflexions sur la théorie cinétique des gaz, J. Phys. théoret. et appl., sér. 4, 1906, vol. 5, pp. 369–403. [19] Kozlov, V.V., Kinetics of Collisionless Continuous Medium, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 235–251. · Zbl 1006.82011 [20] Kozlov, V.V., Notes on Diffusion in Collisionless Medium, Regul. Chaotic Dyn., 2004, vol. 9, no. 1, pp. 29–34. · Zbl 1047.82018 [21] Pokhozhaev, S. I., On Stationary Solutions of the Vlasov-Poisson Equations, Differ. Uravn., 2010, vol. 46, no. 4, pp. 527–534 [Differ. Equ., 2010, vol. 46, no. 4, pp. 530–537]. [22] Batt, J., Faltenbacher, W., and Horst, E., Stationary Spherically Symmetric Models in Stellar Dynamics, Arch. Ration. Mech. Anal., 1986, vol. 93, no. 2, pp. 159–183. · Zbl 0605.70008 [23] Batt, J., Berestycki, H., Decond, P., and Pertname, B., Some Families of Solutions of the Vlasov-Poisson System, Arch. Ration. Mech. Anal., 1988, vol. 104, no. 1, pp. 79–103. · Zbl 0703.35171 [24] Vedenyapin, V.V., On the Classification of Steady-State Solutions of Vlasov’s Equation on the Torus, and a Boundary Value Problem Dokl. Ross. Akad. Nauk, 1992, vol. 323, no. 6, pp. 1004–1006 [Russ. Acad. Sci., Dokl. Math., 1992, vol. 45, no. 2, pp. 459–462]. [25] Carleman, T., Problèmes mathématiques dans la théorie cinétique des gaz, Upsala: Almqvist & Wiksells, 1957. [26] Poincaré, H., Les méthodes nouvelles de la Mécanique céleste: T. 1, Paris: Gauthier-Villars, 1892. [27] Kozlov, V.V., Symmetries, Topology, and Resonances in Hamiltonian Mechanics, Berlin: Springer, 1995. · Zbl 0921.58029 [28] Belotserkovskii, O. M., Fimin, N.N., and Chechetkin, V. M., Application of the Kac Equation to Turbulence Simulation, Zh. Vychisl. Mat. Mat. Fiz., 2010, vol. 50, no. 3, pp. 575–584 [Comput. Math. Math. Phys., 2010, vol. 50, no. 3, pp. 549–557. · Zbl 1224.76070 [29] Pfaffelmoser, K., Global Classical Solutions of the Vlasov-Poisson System in Three Dimensions for General Initial Data, J. Differential Equations, 1992, vol. 95, no. 2, pp. 281–303. · Zbl 0810.35089 [30] Lions, P. L. and Perthame, B., Propagation of Moments and Regularity for the 3-Dimensional Vlasov-Poisson System, Invent. Math., 1991, vol. 105, no. 2, pp. 415–430. · Zbl 0741.35061 [31] Fridman, A. M., Prediction and Discovery of New Structures in Spiral Galaxies, Uspekhi Fiz. Nauk, 2007, vol. 177, no. 2, pp. 121–148 [Physics-Uspekhi, 2007, vol. 50, no. 2, pp. 115–139].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.