# American Institute of Mathematical Sciences

ISSN:
1937-5093

eISSN:
1937-5077

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## Kinetic & Related Models

March 2012 , Volume 5 , Issue 1

Issue on fast reaction - slow diffusion scenarios:
PDE approximations and free boundaries

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2012, 5(1): i-i doi: 10.3934/krm.2012.5.1i +[Abstract](1570) +[PDF](21.3KB)
Abstract:
From this year, Professor Seiji Ukai will step down from the managing editorial board of KRM. Seiji Ukai is a world leading expert in the ﬁeld of mathematical theories of kinetic equations. He is the ﬁrst one who proved the global existence of solutions to the space inhomogeneous Boltzmann equation in 1974, and he has been making important contributions to this area, including his series of recent works on the Boltzmann equation without angular cutoff. We are deeply indebted to the contributions that Seiji made to the area and the journal and wish him all the best in the future.

2012, 5(1): 1-19 doi: 10.3934/krm.2012.5.1 +[Abstract](1924) +[PDF](405.2KB)
Abstract:
We consider the discrete Boltzmann equation for binary gas mixtures. Some known results for half-space problems and shock profile solutions of the discrete Boltzmann for single-component gases are extended to the case of two-component gases. These results include well-posedness results for half-space problems for the linearized discrete Boltzmann equation, existence results for half-space problems for the weakly non-linear discrete Boltzmann equation, and existence results for shock profile solutions of the discrete Boltzmann equation. A characteristic number, corresponding to the speed of sound in the continuous case, is calculated for a class of symmetric models. Some explicit calculations are also made for a simplified 6 + 4 -velocity model.
2012, 5(1): 21-50 doi: 10.3934/krm.2012.5.21 +[Abstract](1399) +[PDF](507.1KB)
Abstract:
This paper studies the non linear Boltzmann equation for a two component gas at the small Knudsen number regime. The solution is found from a truncated Hilbert expansion. The first order of the fluid equations shows the ghost effect. The fluid system is solved when the boundary conditions are close enough to each other. Next the boundary conditions for the kinetic system are satisfied by adding for the first and the second order terms of the expansion Knudsen terms. The construction of such boundary layers requires the study of a Milne problem for mixtures. In a last part the rest term of the expansion is rigorously controled by using a new decomposition into a low and a high velocity part.
2012, 5(1): 51-95 doi: 10.3934/krm.2012.5.51 +[Abstract](3161) +[PDF](1412.8KB)
Abstract:
We propose a PDE chemotaxis model, which can be viewed as a regularization of the Patlak-Keller-Segel (PKS) system. Our modification is based on a fundamental physical property of the chemotactic flux function---its boundedness. This means that the cell velocity is proportional to the magnitude of the chemoattractant gradient only when the latter is small, while when the chemoattractant gradient tends to infinity the cell velocity saturates. Unlike the original PKS system, the solutions of the modified model do not blow up in either finite or infinite time in any number of spatial dimensions, thus making it possible to use bounded spiky steady states to model cell aggregation. After obtaining local and global existence results, we use the local and global bifurcation theories to show the existence of one-dimensional spiky steady states; we also study the stability of bifurcating steady states. Finally, we numerically verify these analytical results, and then demonstrate that solutions of the two-dimensional model with nonlinear saturated chemotactic flux function typically develop very complicated spiky structures.
2012, 5(1): 97-112 doi: 10.3934/krm.2012.5.97 +[Abstract](2144) +[PDF](1764.9KB)
Abstract:
In this paper we develop an improved three dimensional stochastic model for the lay-down of fibers on a moving conveyor belt in the production process of nonwoven materials. The model removes a drawback of a previous 3D model, that is the non-smoothness of the fiber paths. A similar result in the 2D case has been presented in [12]. The resulting equations are investigated for different limit situations and numerical simulations are presented.
2012, 5(1): 113-128 doi: 10.3934/krm.2012.5.113 +[Abstract](1632) +[PDF](422.1KB)
Abstract:
The spectral analysis of a dissipative linear transport operator with a polynomial collision integral by the Szőkefalvi-Nagy - Foiaş functional model is given. An exact estimate for the remainder in the asymptotic of the corresponding evolution semigroup is proved in the isotropic case. In the general case, it is shown that the operator has at most finitely many eigenvalues and spectral singularities and an absolutely continuous essential spectrum. An upper estimate for the remainder is established.
2012, 5(1): 129-153 doi: 10.3934/krm.2012.5.129 +[Abstract](1959) +[PDF](384.4KB)
Abstract:
A collisionless plasma is modeled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge, which is independent of time and space, is assumed. The situation in which mobile negative ions balance the positive charge as $|x|\to\infty$ is considered. Hence, the total positive charge and the total negative charge are both infinite. It is shown, in three spatial dimensions, that smooth solutions may be continued as long as the velocity support remains finite. Also, in the case of spherical symmetry, a bound on velocity support is obtained and hence solutions exist globally in time.
2012, 5(1): 155-184 doi: 10.3934/krm.2012.5.155 +[Abstract](1815) +[PDF](1376.1KB)
Abstract:
Standard hyperbolic solvers for the compressible Euler equations cause increasing approximation errors and have severe stability requirement in the low Mach number regime. It is desired to design numerical schemes that are suitable for all Mach numbers. A second order in both space and time all speed method is developed in this paper, which is an improvement of the semi-implicit framework proposed in [5].
The second order time discretization is based on second order Runge-Kutta method combined with Crank-Nicolson with some implicit terms. This semi-discrete framework is crucial to obtain second order convergence, as well as maintain the asymptotic preserving (AP) property. The AP property indicates that the right limit can be captured in the low Mach number regime. For the space discretization, the pressure term in the momentum equation is divided into two parts. Two subsystems are formed correspondingly, each using different space discretizations. One is discretized by Kurganov-Tadmor central scheme (KT), while the other one is reformulated into an elliptic equation. The proper subsystem division varies with time and the scheme becomes explicit when the time step is small enough.
Compared with previous semi-implicit method, this framework is simpler and natural, with only two linear elliptic equations needed to be solved for each time step. It maintains the AP property of the first order method in [5], improves accuracy and reduces the diffusivity significantly.
2012, 5(1): 185-201 doi: 10.3934/krm.2012.5.185 +[Abstract](1594) +[PDF](397.1KB)
Abstract:
The regularized 13-moment equations (R13) are a successful macroscopic model to describe non-equilibrium gas flows in rarefied or micro situations. Even though the equations have been derived for the nonlinear case and many examples demonstrate the usefulness of the equations, sofar, the important property of an accompanying entropy law could only be shown for the linearized equations [Struchtrup&Torrilhon, Phys. Rev. Lett. 99, (2007), 014502]. Based on an approach suggested by Öttinger [Phys. Rev. Lett. 104, (2010), 120601], this paper presents a nonlinear entropy law for the R13 system. In the derivation the variables and equations of the R13 system are nonlinearily extended such that an entropy law with non-negative production can be formulated. It is then demonstrated that the original R13 system is included in the new equations.
2012, 5(1): 203-221 doi: 10.3934/krm.2012.5.203 +[Abstract](1446) +[PDF](473.3KB)
Abstract:
We derive some fluid-dynamic models for electron transport near a Dirac point in graphene. We start from a kinetic model constituted by a set of spinorial Wigner equations, we make suitable scalings (hydrodynamic or diffusive) of the model and we build moment equations, which we close through a minimum entropy principle. In order to do this we make some assumptions: the usual semiclassical approximation (ħ $\ll 1$), and two further hypothesis, namely Low Scaled Fermi Speed (LSFS) and Strongly Mixed State (SMS), which allow us to explicitly compute the closure.

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