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Volume 1, 2008

Kinetic & Related Models

2008 , Volume 1 , Issue 2

Guest Editors
Boris Belinskiy, Kunquan Lan, Xin Lu, Alain Miranville, and R. Shivaji

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A maximum entropy principle based closure method for macro-micro models of polymeric materials
Yunkyong Hyon, José A. Carrillo, Qiang Du and Chun Liu
2008, 1(2): 171-184 doi: 10.3934/krm.2008.1.171 +[Abstract](735) +[PDF](817.5KB)
Abstract:
We consider the finite extensible nonlinear elasticity (FENE) dumbbell model in viscoelastic polymeric fluids. We employ the maximum entropy principle for FENE model to obtain the solution which maximizes the entropy of FENE model in stationary situations. Then we approximate the maximum entropy solution using the second order terms in microscopic configuration field to get an probability density function (PDF). The approximated PDF gives a solution to avoid the difficulties caused by the nonlinearity of FENE model. We perform the moment-closure approximation procedure with the PDF approximated from the maximum entropy solution, and compute the induced macroscopic stresses. We also show that the moment-closure system satisfies the energy dissipation law. Finally, we show some numerical simulations to verify the PDF and moment-closure system.
Problem of evaporation-condensation for a two component gas in the slab
Stéphane Brull
2008, 1(2): 185-221 doi: 10.3934/krm.2008.1.185 +[Abstract](430) +[PDF](398.1KB)
Abstract:
This paper studies the non linear Boltzmann equation for a two component gas in the situation of hard spheres. A Hilbert expansion of the solution is performed. The first order of the fluid equations shows the ghost effect. The fluid system is solved when the boundary conditions are close to each other.
    The boundary conditions for the kinetic system are satisfied by adding for the first and the second order Knudsen layers. In a last part the rest term is rigorously controled by using a decomposition into a low part velocity and a high part velocity. This constitutes a generalization to the case of a two component gas of the results presented in [15,16].
Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models
Bertrand Lods, Clément Mouhot and Giuseppe Toscani
2008, 1(2): 223-248 doi: 10.3934/krm.2008.1.223 +[Abstract](576) +[PDF](325.1KB)
Abstract:
We consider the linear dissipative Boltzmann equation describing inelastic interactions of particles with a fixed background. For the simplified model of Maxwell molecules first, we give a complete spectral analysis, and deduce from it the optimal rate of exponential convergence to equilibrium. Moreover we show the convergence to the heat equation in the diffusive limit and compute explicitely the diffusivity. Then for the physical model of hard spheres we use a suitable entropy functional for which we prove explicit inequality between the relative entropy and the production of entropy to get exponential convergence to equilibrium with explicit rate. The proof is based on inequalities between the entropy production functional for hard spheres and Maxwell molecules. Mathematical proof of the convergence to some heat equation in the diffusive limit is also given. From the last two points we deduce the first explicit estimates on the diffusive coefficient in the Fick's law for (inelastic hard-spheres) dissipative gases.
Towards a mathematical theory of complex socio-economical systems by functional subsystems representation
Giulia Ajmone Marsan, Nicola Bellomo and Massimo Egidi
2008, 1(2): 249-278 doi: 10.3934/krm.2008.1.249 +[Abstract](699) +[PDF](329.0KB)
Abstract:
This paper deals with the development of a mathematical theory for complex socio-economical systems. The approach is based on the methods of the mathematical kinetic theory for active particles, which describes the evolution of large systems of interacting entities which are carriers of specific functions, in our case economical activities. The method is implemented with the concept of functional subsystems constituted by aggregated entities which have the ability of expressing socio-economical purposes and functions.
Modelling and simulation of vehicular traffic jam formation
Pierre Degond and Marcello Delitala
2008, 1(2): 279-293 doi: 10.3934/krm.2008.1.279 +[Abstract](616) +[PDF](195.5KB)
Abstract:
This paper deals with the modelling and simulation of traffic flow phenomena at the macroscopic level, based on a suitable development of the Aw-Rascle model, [2], and its modification, [4], [5]. An acceleration term and a minimal velocity--dependent safety distance are introduced in the evolution equations. Then, an eulerian computational scheme is introduced to simulate the formation and evolution of jams. The results are compared with those obtained in [4], where a lagrangian computational method was used instead.
Qualitative analysis of the generalized Burnett equations and applications to half--space problems
Marzia Bisi, Maria Paola Cassinari and Maria Groppi
2008, 1(2): 295-312 doi: 10.3934/krm.2008.1.295 +[Abstract](383) +[PDF](831.4KB)
Abstract:
The Generalized Burnett Equations, very recently introduced by Bobylev [3,4], are tested versus Fluid--Dynamic applications, considering the classical steady evaporation/condensation problem. By means of the methods of the qualitative theory of dynamical systems, comparison is made to other kinetic and hydrodynamic models, and indications on an appropriate choice of the disposable parameters are obtained.
The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients
Quansen Jiu and Zhouping Xin
2008, 1(2): 313-330 doi: 10.3934/krm.2008.1.313 +[Abstract](922) +[PDF](247.8KB)
Abstract:
This paper concerns with Cauchy problems for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficients. Two cases are considered here, first, the initial density is assumed to be integrable on the whole real line. Second, the deviation of the initial density from a positive constant density is integrable on the whole real line. It is proved that for both cases, weak solutions for the Cauchy problem exist globally in time and the large time asymptotic behavior of such weak solutions are studied. In particular, for the second case, the phenomena of vanishing of vacuum and blow-up of the solutions are presented, and it is also shown that after the vanishing of vacuum states, the globally weak solution becomes a unique strong one. The initial vacuum is permitted and the results apply to the one-dimensional Saint-Venant model for shallow water.

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