On a kinetic model for shallow water waves

The system of shallow water waves is one of the classical examples for non-linear, two-dimensional conservation laws. The paper investigates a simple kinetic equation depending on a parameter ? which leads for ? → 0 to the system of shallow water waves. The corresponding ‘equilibrium’ distribution function has a compact support which depends on the eigenvalues of the hyperbolic system. It is shown that this kind of kinetic approach is restricted to a special class of non-linear conservation laws. The kinetic model is used to develop a simple particle method for the numerical solution of shallow water waves. The particle method can be implemented in a straightforward way and produces in test examples sufficiently accurate results.     

The realization of unilateral constraints in mechanical systems with kinetic energy decay

A method of realizing unilateral constraints by introducing a small parameter into the kinetic energy, such that the kinetic energy of the system degenerates on a certain manifold in configuration space when the parameter is equal to zero, is considered. The behaviour of the system in the neighbourhood of the degeneracy manifold in the multidimensional case when the small parameter approaches zero is investigated. The results obtained are used in the problem of the motion of a double mathematical pendulum when the mass of the point closest to the suspension point is small.     

Modified method of splitting with respect to physical processes for solving radiation gas dynamics equations

An approach based on a modified splitting method is proposed for solving the radiation gas dynamics equations in the multigroup kinetic approximation. The idea of the approach is that the original system of equations is split using the thermal radiation transfer equation rather than the energy equation. As a result, analytical methods can be used to solve integrodifferential equations and problems can be computed in the multigroup kinetic approximation without iteration with respect to the collision integral or matrix inversion. Moreover, the approach can naturally be extended to multidimensional problems. A high-order accurate difference scheme is constructed using an approximate Godunov solver for the Riemann problem in two-temperature gas dynamics.     

Qualitative response of interaction networks: application to the validation of biological models

We advocate the use of qualitative models for the analysis of shift equilibria in large biological systems. We present a mathematical method, allowing qualitative predictions to be made of the behaviour of a biological system. These predictions are not dependent on specific values of the kinetic constants. We show how these methods can be used to improve understanding of a complex regulatory system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

On the construction and investigation of solutions of boundary value problems of thermoviscoelasticity

The uncoupled mixed boundary value problem of thermoviscoelasticity is considered in a quasistatic formulation. The temperature distribution is assumed nonstationary and inhomogeneous. The influence of the temperature on the viscoelastic properties of the material is taken into account by the introduction of a reduced time. The equations of state of the material are written in differential form as a system of kinetic equations in some tensor-type strain parameters. The system mentioned is equivalent to a Volterra integral equation with kernel in the form of a sum of exponents. The differential approach used is apparently more convenient for numerical realization /1/ (especially in nonuniform problems) and results in a substantially different mathematical formulation as compared with that based on the integral form of writing the equations of state investigated in /2,3/. Precisely for going over to the boundary value problem are the kinetic differential equations converted into an operator differential equation in Hubert space. The existence, uniqueness, and stability of the solution of the problem formulated are established, and conditions for the convergence of the Galerkin approximations and the stability of the difference approximations in time are formulated.     

A kinetic scheme for unsteady pressurised flows in closed water pipes

The aim of this paper is to present a kinetic numerical scheme for the computations of transient pressurised flows in closed water pipes. Firstly, we detail the mathematical model written as a conservative hyperbolic partial differentiel system of equations, and then we recall how to obtain the corresponding kinetic formulation. Then we build the kinetic scheme ensuring an upwinding of the source term due to the topography performed in a close manner described by Perthame and Simeoni (2001) [1] and Botchorishvili et al. (2003) [2] using an energetic balance at microscopic level. The validation is lastly performed in the case of a water hammer in an uniform pipe: we compare the numerical results provided by an industrial code used at EDF-CIH (France), which solves the Allievi equation (the commonly used equation for pressurised flows in pipes) by the method of characteristics, with those of the kinetic scheme. It appears that they are in a very good agreement.     

A mean-field approach for the asymptotic tracking problem of moving continuum target clouds

We propose a new coupled kinetic system arising from the asymptotic tracking of a continuum target cloud, and study its asymptotic tracking property. For the proposed kinetic system, we present an energy functional which is monotonic and distance between particle trajectories corresponding to kinetic equations for target, and tracking ensembles tend to zero asymptotically under a suitable sufficient framework. The framework is formulated in terms of system parameters and initial data.     

Nonlinear nonequilibrium kinetic model of the boltzmann equation for monatomic gases

A model kinetic equation approximating the Boltzmann equation in a wide range of nonequilibrium gas states was constructed to describe rarefied gas flows. The kinetic model was based on a distribution function depending on the absolute velocity of the gas particles. Highly efficient in numerical computations, the model kinetic equation was used to compute a shock wave structure. The numerical results were compared with experimental data for argon.     

Dynamics on rugged landscapes of energy and ultrametric diffusion

We discuss the interbasin kinetics approximation for random walk on a complex (rugged) landscape of energy. In this approximation the random walk is described by the system of kinetic equations corresponding to transitions between the local minima of energy. If we approximate the transition rates between the local minima by the Arrhenius formula then the system of kinetic equations will be hierarchical. We discuss for a generic landscape of energy the anzats of interbasin kinetics which is equivalent to the ultrametric diffusion generated by an ultrametric pseudodifferential operator.     

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.     

Computation of dynamical phase transitions in solids

This paper deals with the dynamics of phase boundaries in a nonlinear elastic two-phase material. We consider the elasticity system in 1D and the equations of anti-plane shear motion in 2D, where effects of viscosity and capillarity are neglected. These first-order conservation laws allow to represent phase boundaries as shock-like sharp interfaces. However, in contrast to what is known for homogeneous materials, the entropy inequality does not select a unique solution, and an additional criterion, the so-called kinetic relation, is required.Based on a scheme introduced by Hou, Rosakis and LeFloch [T. Hou, Ph. Rosakis, P.G. LeFloch, A level-set approach to the computation of twinning and phase-transition dynamics, J. Comput. Phys. 150 (1999) 302–331] we focus on the numerical solution of a specific model system. Using a level-set technique to enforce the kinetic relation on the discrete level leads to a reformulation of the original system in the form of a system of conservation laws coupled to a Hamilton–Jacobi equation for each phase boundary. The numerical method for the reformulated system is constructed for unstructured meshes (in 2D), and a self-adaptive algorithm is introduced.In the 1D-case we show that the reformulated system has a solution that corresponds to exact dynamical phase boundaries of the elasticity system which obey the kinetic relation. To validate the method in 2D, we present computations on the interaction of a plane wave with a phase boundary. The efficiency of the adaptation mechanism is demonstrated by an example showing the development of microstructures by twinning.     

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.

     

Cross-diffusion induced instability and stability in reaction-diffusion systems

In a reaction-diffusion system, diffusion can induce the instability of a uniform equilibrium  which is stable with respect to a constant perturbation, as shown by Turing in 1950s. We show that cross-diffusion can destabilize  a uniform equilibrium  which is stable for the kinetic and self-diffusion reaction systems; on the other hand, cross-diffusion can also stabilize  a uniform equilibrium which is stable for the kinetic system but unstable for the self-diffusion reaction system. Application is given to predator-prey system with preytaxis and vegetation pattern formation in a water-limited ecosystem.     

Coagulation, diffusion and the continuous Smoluchowski equation

The Smoluchowski equations are a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers or by positive reals, these corresponding to the discrete or the continuous form of the equations. For dimension d≥3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a method similar to that used to derive the discrete form of the equations in [A. Hammond, F. Rezakhanlou, The kinetic limit of a system of coagulating Brownian particles, Arch. Ration. Mech. Anal. 185 (2007) 1–67]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of the cited work. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the equations.     

Seebeck Effect in Silicon Semiconductors

An extended hydrodynamic model will be used for the coupled system of electrons and phonons. This system is formed by a set of balance equations derived from the Bloch-Boltzmann-Peierls (BBP) kinetic equations applying the moment method and solving the problem of the closure by means of the Maximum Entropy Principle of Extended Thermodynamics. By using this model with a suitable limit, thermoelectric effects in bulk silicon are investigated.     

Nonequilibrium jet boundary layer in a polyatomic gas

The two-dimensional nonequilibrium hypersonic free jet boundary layer gas flow in the near wake of a body is studied using a closed system of macroscopic equations obtained (as a thin-layer version) from moment equations of kinetic origin for a polyatomic single-component gas with internal degrees of freedom. (This model is can be used to study flows with strong violations of equilibrium with respect to translational and internal degrees of freedom.) The solution of the problem under study (i.e., the kinetic model of a nonequilibrium homogeneous polyatomic gas flow in a free jet boundary layer) is shown to be related to the known solution of the well-studied simpler problem of a Navier-Stokes free jet boundary layer, and a method based on this relation is proposed for solving the former problem. It is established that the gas flow velocity distribution along the separating streamline in the kinetic problem of a free jet boundary layer coincides with the distribution obtained by solving the Navier-Stokes version of the problem. It is found that allowance for the nonequilibrium nature of the flow with respect to the internal and translational degrees of freedom of a single-component polyatomic gas in a hypersonic free jet boundary layer has no effect on the base pressure and the wake angle.     

Physical model,theoretical aspects and applications of the flight of a ball in the atmosphere. Part I: Modelling of forces and torque,and theoretical prospects

A model of the forces and the torque operating on a ball that is flying with rotation in the atmosphere of the Earth, and the resulting system of ordinary differential equations, are derived from mechanics and aerodynamics. The system of equations allows the theoretical aspects of the flight of a ball, such as the boundedness of its kinetic energy, the curvature of the orbit or the velocity function, to be investigated using certain transformations of the variables. The solutions of the corresponding ordinary or boundary value problems, computed numerically, are used to treat certain tasks in international ball games, for example, the maximum and minimum velocities of a volleyball service.     

Quantum kinetic equations of the system “graphene + dimensionally quantized film”

We consider the problem of epitaxial graphene formed on a dimensionally quantized film. We give a microscopic derivation of the quantum kinetic equations for the system ??graphene + dimensionally quantized film.?? Based on the obtained equations, we investigate the features of electron states of the system under consideration and obtain analytic expressions for the transferred charge in this system.