# American Institute of Mathematical Sciences

ISSN:
1937-5093

eISSN:
1937-5077

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## Kinetic & Related Models

September 2013 , Volume 6 , Issue 3

Issue on Evolution Equations and Mathematical Models in the Applied Sciences

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2013, 6(3): 459-479 doi: 10.3934/krm.2013.6.459 +[Abstract](2099) +[PDF](702.8KB)
Abstract:
This paper deals with the modeling of social competition, possibly resulting in the onset of extreme conflicts. More precisely, we discuss models describing the interplay between individual competition for wealth distribution that, when coupled with political stances coming from support or opposition to a Government, may give rise to strongly self-enhanced effects. The latter may be thought of as the early stages of massive unpredictable events known as Black Swans, although no analysis of any fully-developed Black Swan is provided here. Our approach makes use of the framework of the kinetic theory for active particles, where nonlinear interactions among subjects are modeled according to game-theoretical principles.
2013, 6(3): 481-503 doi: 10.3934/krm.2013.6.481 +[Abstract](1624) +[PDF](499.9KB)
Abstract:
We are concerned with global existence and large-time behavior of solutions to the isentropic electric-magnetohydrodynamic equations in a bounded domain $\Omega\subseteq\mathbb{R}^{N}$, $N=2,\ 3$. We establish the existence and large-time behavior of global weak solutions through a three-level approximation, energy estimates on condition that the adiabatic constant satisfies $\gamma>3/2$.
2013, 6(3): 505-532 doi: 10.3934/krm.2013.6.505 +[Abstract](1639) +[PDF](1238.2KB)
Abstract:
The Bloch decomposition plays a fundamental role in the study of quantum mechanics and wave propagation in periodic media. Most of the homogenization theory developed for the study of high frequency or semi-classical limit for these problems assumes no crossing of the Bloch bands, resulting in a classical Liouville equation in the limit along each Bloch band.
In this article, we derive semi-classical models for the Schrödinger equation in periodic media that take into account band crossings, which is important to describe quantum transitions between Bloch bands. Our idea is still based on the Wigner transform (on the Bloch eigenfunctions), but in taking the semi-classical approximation, we retain the off-diagonal entries of the Wigner matrix, which cannot be ignored near the points of band crossings. This results in coupled inhomogeneous Liouville systems that can suitably describe quantum tunneling between bands that are not well-separated. We also develop a domain decomposition method that couples these semi-classical models with the classical Liouville equations (valid away from zones of band crossings) for a multiscale computation. Solutions of these models are numerically compared with those of the Schrödinger equation to justify the validity of these new models for band-crossings.
2013, 6(3): 533-543 doi: 10.3934/krm.2013.6.533 +[Abstract](1541) +[PDF](324.0KB)
Abstract:
A brief derivation of a specific regularization for the magnetic gas dynamic system of equations is given in the case of general equations of gas state (in presence of a body force and a heat source). The entropy balance equation in two forms is also derived for the system. For a constant magnetic regularization parameter and under a standard condition on the heat source, we show that the entropy production rate is nonnegative.
2013, 6(3): 545-556 doi: 10.3934/krm.2013.6.545 +[Abstract](1643) +[PDF](357.1KB)
Abstract:
In this paper, logarithmically improved regularity criteria for the generalized Navier-Stokes equations are established in terms of the velocity, vorticity and pressure, respectively. Here $BMO$, the Triebel-Lizorkin and Besov spaces are used, which extend usual Sobolev spaces much. Similar results for the quasi-geostrophic flows and the generalized MHD equations are also listed.
2013, 6(3): 557-587 doi: 10.3934/krm.2013.6.557 +[Abstract](1406) +[PDF](835.4KB)
Abstract:
We derive a hierarchy of closures based on perturbations of well-known entropy-based closures; we therefore refer to them as perturbed entropy-based models. Our derivation reveals final equations containing an additional convective and diffusive term which are added to the flux term of the standard closure. We present numerical simulations for the simplest member of the hierarchy, the perturbed $M_1$ or $PM_1$ model, in one spatial dimension. Simulations are performed using a Runge-Kutta discontinuous Galerkin method with special limiters that guarantee the realizability of the moment variables and the positivity of the material temperature. Improvements to the standard $M_1$ model are observed in cases where unphysical shocks develop in the $M_1$ model.
2013, 6(3): 589-599 doi: 10.3934/krm.2013.6.589 +[Abstract](1869) +[PDF](314.2KB)
Abstract:
In this article, the uniqueness of weak solutions to the continuous coagulation and multiple fragmentation equation is proved for a large range of unbounded coagulation and multiple fragmentation kernels. The multiple fragmentation kernels may have a singularity at origin. This work generalizes the preceding ones, by including some physically relevant coagulation and fragmentation kernels which were not considered before.
2013, 6(3): 601-623 doi: 10.3934/krm.2013.6.601 +[Abstract](1560) +[PDF](421.3KB)
Abstract:
In this paper a half space problem for the one-dimensional Landau equation with specular reflective boundary condition is investigated. We show that the solution to the Landau equation converges to a global Maxwellian. Moreover, a time-decay rate is also obtained.
2013, 6(3): 625-648 doi: 10.3934/krm.2013.6.625 +[Abstract](1463) +[PDF](475.5KB)
Abstract:
We prove that the linearized non-cutoff Boltzmann operator with Maxwellian molecules is exactly equal to a fractional power of the linearized Landau operator which is the sum of the harmonic oscillator and the spherical Laplacian. This result allows to display explicit sharp coercive estimates satisfied by the linearized non-cutoff Boltzmann operator for both Maxwellian and non-Maxwellian molecules.
2013, 6(3): 649-670 doi: 10.3934/krm.2013.6.649 +[Abstract](1657) +[PDF](463.0KB)
Abstract:
This paper is concerned with nonlinear stability of viscous shock profiles for the one-dimensional isentropic compressible Navier-Stokes equations. For the case when the diffusion wave introduced in [6, 7] is excluded, such a problem has been studied in [5, 11] and local stability of weak viscous shock profiles is well-established, but for the corresponding result with large initial perturbation, fewer results have been obtained. Our main purpose is to deduce the corresponding nonlinear stability result with large initial perturbation by exploiting the elementary energy method. As a first step toward this goal, we show in this paper that for certain class of large" initial perturbation which can allow the initial density to have large oscillation, similar stability result still holds. Our analysis is based on the continuation argument and the technique developed by Kanel' in [4].

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