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Kinetic & Related Models

2010 , Volume 3 , Issue 3

Special issue
on evolutionary PDEs in fluid mechanics

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Galerkin methods for primary ion transport in inhomogeneous media
Mohammad Asadzadeh, Anders Brahme and Jiping Xin
2010, 3(3): 373-394 doi: 10.3934/krm.2010.3.373 +[Abstract](292) +[PDF](339.1KB)
Abstract:
This paper concerns the energy deposition of high-energy (e.g., $\approx 50-500$ MeV) proton and carbon ions and high-energy electrons (of $\approx 50$ MeV), in inhomogeneous media. Our goal is to develop a flexible model incorporated with the analytic theory for ions based on bipartition and Fokker-Planck developments. Both procedures are leading to convection dominated convection diffusion equations. We study convergence for semi-discrete and fully discrete approximations of a such obtained equation, for a broad beam model, using the standard Galerkin and streamline diffusion finite element methods. The analytic broad beam model of the light ion absorbed dose were compared with the results of the modified Monte Carlo (MC) code SHIELD-HIT+ and those of Galerkin streamline diffusion approach.
On a generalized Boltzmann equation for non-classical particle transport
Martin Frank and Thierry Goudon
2010, 3(3): 395-407 doi: 10.3934/krm.2010.3.395 +[Abstract](296) +[PDF](161.3KB)
Abstract:
We are interested in non-standard transport equations where the description of the scattering events involves an additional "memory variable''. We establish the well posedness and investigate the diffusion asymptotics of such models. While the questions we address are quite classical the analysis is original since the usual dissipative properties of collisional transport equations is broken by the introduction of the memory terms.
Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary
Feimin Huang, Xiaoding Shi and Yi Wang
2010, 3(3): 409-425 doi: 10.3934/krm.2010.3.409 +[Abstract](387) +[PDF](200.0KB)
Abstract:
A free boundary problem for the one-dimensional compressible Navier-Stokes equations in Eulerian coordinate is investigated. The stability of the viscous shock wave to the free boundary problem is established under some smallness conditions. The proof is given by an elementary energy method.
Asymptotic behaviour of reversible chemical reaction-diffusion equations
Ivan Gentil and Bogusław Zegarlinski
2010, 3(3): 427-444 doi: 10.3934/krm.2010.3.427 +[Abstract](312) +[PDF](233.3KB)
Abstract:
We investigate the asymptotic behavior of a large class of reversible chemical reaction-diffusion equations with the same diffusion. In particular we prove the optimal rate in two cases : when there is no diffusion and in the classical "two-by-two" case.
The uniformly heated inelastic Boltzmann equation in Fourier space
Ralf Kirsch and Sergej Rjasanow
2010, 3(3): 445-456 doi: 10.3934/krm.2010.3.445 +[Abstract](275) +[PDF](170.8KB)
Abstract:
In this article, we present an alternative formulation of the Boltzmann equation for diffusively driven granular media. The equation is considered with minimal a priori assumptions, i.e. in weak form in the sense of tempered distributions. Using shifted test functions and the Fourier transform, it is seen that the transformed problem contains only a threefold integral. For constant restitution coefficients and the variable hard spheres model, explicit expressions of the integral kernel in the transformed collision operator are obtained. The version of the equation derived here is a true extension of the elastic case. Some well-known results for Maxwell molecules with inelastic interactions are recovered.
Strengthened convergence of marginals to the cubic nonlinear Schrödinger equation
Alessandro Michelangeli
2010, 3(3): 457-471 doi: 10.3934/krm.2010.3.457 +[Abstract](225) +[PDF](221.9KB)
Abstract:
We rewrite a recent derivation of the cubic non-linear Schrödinger equation by Adami, Golse, and Teta in the more natural form of the asymptotic factorisation of marginals at any fixed time and in the trace norm. This is the standard form in which the emergence of the non-linear effective dynamics of a large system of interacting bosons is proved in the literature.
On permanent regimes for non-autonomous linear evolution equations in Banach spaces with applications to transport theory
Mustapha Mokhtar-Kharroubi
2010, 3(3): 473-499 doi: 10.3934/krm.2010.3.473 +[Abstract](210) +[PDF](289.4KB)
Abstract:
This paper deals with existence and uniqueness of permanent (i.e. defined for all time $t\in \mathbb{R}$) solutions of non-autonomous linear evolution equations governed by strongly stable (at $-\infty $) evolution families in Banach spaces and driven by permanent bounded forcing terms. In particular, we study the existence and uniqueness of (asymptotically) almost-periodic solutions driven by (asymptotically) almost-periodic forcing terms. Systematic applications to some non-autonomous linear kinetic equations in arbitrary geometries relying on their dispersive properties are given.
Numerical simulation of a kinetic model for chemotaxis
Nicolas Vauchelet
2010, 3(3): 501-528 doi: 10.3934/krm.2010.3.501 +[Abstract](315) +[PDF](593.1KB)
Abstract:
This paper is devoted to numerical simulations of a kinetic model describing chemotaxis. This kinetic framework has been investigated since the 80's when experimental observations have shown that the motion of bacteria is due to the alternance of 'runs and tumbles'. Since parabolic and hyperbolic models do not take into account the microscopic movement of individual cells, kinetic models have become of a great interest. Dolak and Schmeiser (2005) have then proposed a kinetic model describing the motion of bacteria responding to temporal gradients of chemoattractants along their paths. An existence result for this system is provided and a numerical scheme relying on a semi-Lagrangian method is presented and analyzed. An implementation of this scheme allows to obtain numerical simulations of the model and observe blow-up patterns that differ greatly from the case of Keller-Segel type of models.

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