Analysis of a coupled system of kinetic equations and conservation laws: Rigorous derivation and existence theory via defect measures

In this paper we introduce a coupled system of kinetic equations of B.G.K. type and then study its hydrodynamic limit. We obtain as a consequence the rigorous derivation and existence theory for a coupled system of kinetic equations and their hydrodynamic (conservation laws) limit. The latter is a particular case of the coupled system of Boltzmann and Euler equations. A fundamental element in this study is the rigorous derivation and justification of the interface conditions between the kinetic model and its hydrodynamic conservation laws limit, which is obtained using a new regularity theory introduced herein.

     

The development and substantiation of the kinetic theory of gases in the twentieth century

A new kinetic theory that is close to dynamic processes is constructed. A system of M integro-differential equations and a system of M partial differential equations are obtained. The theory is demonstrated using the examples of the calculation of the structure of intense shock-waves and by calculating turbulent flows in a plane channel. It is shown that the theory of the structure of high-intensity shock-waves agrees with remarkable accuracy with numerous experimental data. Calculations of turbulent flow approximate quite well to experimental data, but it is remarkable that a single theory can describe both the turbulent core at the centre of the channel and the laminar sublayer on the wall so well.     

A phenomenological and kinetic description of diffusion and heat transport in multicomponent gas mixtures and plasma

Different forms of expressing diffusion and heat fluxes in multicomponent mixtures, obtained by methods of non-equilibrium thermodynamics and the kinetic theory of gas mixtures, are analysed and compared. It is shown that an alternative representation of the linear relations of non-equilibrium thermodynamics is possible, which enables them to be written in a form similar to that of the well-known Stefan–Maxwell equations. A relation between the phenomenological coefficients of non-equilibrium thermodynamics and the corresponding transport coefficients obtained in kinetic theory is established, with a confirmation that the Onsager reciprocity relations are satisfied. It is shown that there is an advantage in writing the transport relations on the basis of the “forces in terms of fluxes” representation, compared with the classical “fluxes in terms of forces” representation, used in standard schemes of phenomenological non-equilibrium thermodynamics and the Chapman–Enskog method, traditional for kinetic theory. A generalization of the Stefan-Maxwell equations and the equation for the heat flux is considered, which takes into account the contribution to these equations of the time and space derivatives of the fluxes. The relaxation form of the equations obtained enable one to approach the analysis of the propagation of small heat and concentration perturbations in gas mixtures to be justified, which, within the framework of classical transport relations, propagate with infinitely high velocity. The results presented in this review enable one to determine the areas of effective application of different methods of describing diffusion and heat transfer in multicomponent gas mixtures when solving specific gas-dynamic problems.     

Wiener-Hopf equation: Kernels representable as a superposition of complex-valued exponents

The paper studies the Wiener-Hopf equations with kernels representable as superposition of complex-valued exponents. Such kernels arise in the kinetic gas theory, in the radiation transfer, etc. By application of a special, three-factor expansion of the initial uninvertible operator, the solution of the considered equation is reduced to those of two simple Volterra equations and a Wiener-Hopf integral equation with a contractive operator. A structural existence theorem is proved.     

Mathematical model of a kinetic boundary layer

The two-dimensional (plane) problem of a hypersonic kinetic boundary layer developing on a thin body in the case of a homogeneous polyatomic gas flow with no dissociation or electron excitation is considered assuming that energy exchange between translational and internal molecular degrees of freedom is easy. (The approximation of a hypersonic kinetic boundary layer arises from the kinetic theory of gases and, within the thin-layer model, takes into account the strong nonequilibrium of the hypersonic flow with respect to translational and internal degrees of freedom of the gas particles.) A method is proposed for constructing the solution of the given kinetic problem in terms of a given solution of an equivalent well-studied classical Navier-Stokes hypersonic boundary layer problem (which is traditionally formulated on the basis of the Navier-Stokes equations).     

Evolution of a quantum system of many particles interacting via the generalized Yukawa potential

We study the evolution of a system of N particles that have identical masses and charges and interact via the generalized Yukawa potential. The system is placed in a bounded region. The evolution of such a system is described by the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) chain of quantum kinetic equations. Using semigroup theory, we prove the existence of a unique solution of the BBGKY chain of quantum kinetic equations with the generalized Yukawa potential.     

A general framework for modeling tumor-immune system competition at the mesoscopic level

We propose a class of nonlinear integro-differential equations that at the mesoscopic level models the competition between a tumor and the immune system. The model describes the evolution of a distribution function of the microscopic parameter referred to as activity of cells. The idea is somehow similar to the Enskog theory in kinetic theory. By averaging with respect to the parameter, the mesoscopic class of models reduces to the general class of macroscopic models introduced by A. d’Onofrio that may assess the effect of delays in stimulation of the immune system by tumor cells. The existence and uniqueness theory is developed.     

On the modelling of crowd dynamics by generalized kinetic models

We introduce a new approach for modelling the behavior of pedestrians which borrows techniques from the kinetic theory of rarefied gazes. This approach is based on concepts from the kinetic theory for ‘active particles’. In our modelling of crowd movements, pedestrians are regarded as entities whose microscopic states include a geometric variable (to denote position), a mechanical variable (to denote speed), and a variable to describe the rate of development of individual strategies that influence the collective dynamics. Indeed, the third variable is introduced to account for the ability of individuals to modify their dynamics according to specific objectives and strategies. The equations eventually obtained are a set of nonlinear integro-differential equations of Vlasov or Boltzmann type.     

A dynamical systems approach based on averaging to model the macroscopic flow of freeway traffic

The flow of traffic exhibits distinct characteristics under different conditions, reflecting the congestion during peak hours and relatively free motion during off-peak hours. This requires one to use different mathematical equations to describe the diverse traffic characteristics. Thus, the flow of traffic is best described by a hybrid system, namely different governing equations for the different regimes of response, and it is such a hybrid approach that is investigated in this paper. Existing models for the flow of traffic treat traffic as a continuum or employ techniques similar to those used in the kinetic theory of gases, neither of these approaches gainfully exploit the hybrid nature of the problem. Spurious two-way propagation of disturbances that are physically unacceptable are predicted by continuum models for the flow of traffic. The number of vehicles in a typical section of the highway does not justify its being modeled as a continuum. It is also important to recognize that the basic premises of kinetic theory are not appropriate for the flow of traffic (see [S. Darbha, K.R. Rajagopal, Limit of a collection of dynamical systems: an application to modeling the flow of traffic, Mathematical Models and Methods in Applied Sciences 12 (10) (2002) 1381–1399] for a rationale for the same). A model for the flow of traffic that does not treat traffic as a continuum or use notions from kinetic theory is developed here and corroborated with real-time data collected on US 183 in Austin, Texas. Predictions based on the hybrid system model seem to agree reasonably well with the data collected on US 183.     

Chain of BBGKY equations for bimolecular chemical reactions

A dynamically verified statistical theory of moderately dense gases developed by Bogoliubov and others is generalized to the case of bimolecular chemical reactions in a gas. The corresponding chain of BBGKY equations is derived. From this chain, the kinetic equations for one-molecule distribution functions are obtained in the approximation of bimolecular and trimolecular interactions. Deceased. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 2, pp. 163–178, May, 1997.     

Flammes planes avec transport multi-espèce et chimie complexe

We consider plane laminar flames with multicomponent transport and complex chemistry. The governing equations are derived from the kinetic theory of gases. An arbitrary number of reversible chemical reactions and temperature dependent species specific heats are considered in the model. The most general form for multicomponent transport fluxes given by the kinetic theory is also taken into account. Upon first considering a bounded domain and then letting the size of the domain to go to infinity, we obtain an existence theorem. A priori estimates fundamentally rely on the entropy which correlates the transport fluxes and the gradients.     

Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy and the kinetic theory of dusty plasmas

We consider the modern state of a consistent kinetic theory of dusty plasmas. We present the derivation of equations for microscopic phase densities of plasma particles and grains. Such equations are suitable for extending the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy to the case of dusty plasmas and for deriving the kinetic equations with regard for both elastic and inelastic particle collisions. Moreover, we describe the effective grain-grain potentials kinetically.     

Possibility of explaining the existence of vortexlike hydrodynamic structures based on the theory of stationary kinetic equations

The possibility of describing vortex structures in quasi-one-dimensional plane flows by applying kinetic equations and bifurcation theory is examined. The Lyapunov-Schmidt method is used to obtain a system of Riccati-type generalized bifurcation equations. An analysis of its properties leads to conditions for the existence of vortex structures.     

Reductions of Kinetic Equations to Finite Component Systems

We consider two distinguish approaches for extraction of finite component systems from kinetic equations. The first method is based on the theory of generalized functions, which in simplest case is nothing but the so called multi flow hydrodynamics well known in plasma physics. An alternative is the so called the moment decomposition method successfully utilized for hydrodynamic chains. The method of hydrodynamic reductions successfully utilized in the theory of integrable hydrodynamic chains is applied to the local and nonlocal kinetic equations. N component reductions parameterized by N?1 arbitrary constants for non-hydrodynamic chain arising in the theory of high frequency nonlinear waves in electron plasma are found. These evolution dispersive systems equipped by a local Hamiltonian structure possess periodic solutions.