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Kinetic & Related Models

2015 , Volume 8 , Issue 1

Issue on numerical methods based on homogenization and two-scale convergence

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Convergence rate for the method of moments with linear closure relations
Yves Bourgault, Damien Broizat and Pierre-Emmanuel Jabin
2015, 8(1): 1-27 doi: 10.3934/krm.2015.8.1 +[Abstract](481) +[PDF](475.9KB)
Abstract:
We study linear closure relations for the moments' method applied to simple kinetic equations. The equations are linear collisional models (velocity jump processes) which are well suited to this type of approximation. In this simplified, 1 dimensional setting, we are able to prove stability estimates for the method (with a kinetic interpretation by a BGK model). Moreover we are also able to obtain convergence rates which automatically increase with the smoothness of the initial data.
Time periodic solutions of the non-isentropic compressible fluid models of Korteweg type
Hong Cai, Zhong Tan and Qiuju Xu
2015, 8(1): 29-51 doi: 10.3934/krm.2015.8.29 +[Abstract](420) +[PDF](511.0KB)
Abstract:
In this paper, the non-isentropic compressible Navier-Stokes-Korteweg system with a time periodic external force is considered in $\mathbb{R}^n$. The optimal time decay rates are obtained by spectral analysis. Using the optimal decay estimates, we show that the existence, uniqueness and time-asymptotic stability of time periodic solutions when the space dimension $n\geq 5$. Our proof is based on a combination of the energy method and the contraction mapping theorem.
On the Boltzmann equation with the symmetric stable Lévy process
Yong-Kum Cho
2015, 8(1): 53-77 doi: 10.3934/krm.2015.8.53 +[Abstract](502) +[PDF](476.3KB)
Abstract:
As for the spatially homogeneous Boltzmann equation of Maxwellian molecules with the fractional Fokker-Planck diffusion term, we consider the Cauchy problem for its Fourier-transformed version, which can be viewed as a kinetic model for the stochastic time-evolution of characteristic functions associated with the symmetric stable Lévy process and the Maxwellian collision dynamics. Under a non-cutoff assumption on the kernel, we establish a global existence theorem with maximum growth estimate, uniqueness and stability of solutions.
Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion
Vincent Giovangigli and Wen-An Yong
2015, 8(1): 79-116 doi: 10.3934/krm.2015.8.79 +[Abstract](504) +[PDF](666.5KB)
Abstract:
We analyze a mathematical model for the relaxation of translational and internal temperatures in a nonequilibrium gas. The system of partial differential equations---derived from the kinetic theory of gases---is recast in its natural entropic symmetric form as well as in a convenient hyperbolic-parabolic symmetric form. We investigate the Chapman-Enskog expansion in the fast relaxation limit and establish that the temperature difference becomes asymptotically proportional to the divergence of the velocity field. This asymptotic behavior yields the volume viscosity term of the limiting one-temperature fluid model.
Stability of the stationary solution of the cauchy problem to a semiconductor full hydrodynamic model with recombination-generation rate
Haifeng Hu and Kaijun Zhang
2015, 8(1): 117-151 doi: 10.3934/krm.2015.8.117 +[Abstract](498) +[PDF](531.1KB)
Abstract:
We study the Cauchy problem of a 1-D full hydrodynamic model for semiconductors where the energy equations are included. In the case of recombination-generation effects between electrons and holes being taken into consideration, the existence and uniqueness of a subsonic stationary solution of the related system are established. The convergence of the global smooth solution to the stationary solution exponentially is proved as time tends to infinity.
Global magnetic confinement for the 1.5D Vlasov-Maxwell system
Toan T. Nguyen, Truyen V. Nguyen and Walter A. Strauss
2015, 8(1): 153-168 doi: 10.3934/krm.2015.8.153 +[Abstract](453) +[PDF](462.5KB)
Abstract:
We establish the global-in-time existence and uniqueness of classical solutions to the ``one and one-half'' dimensional relativistic Vlasov--Maxwell systems in a bounded interval, subject to an external magnetic field which is infinitely large at the spatial boundary. We prove that the large external magnetic field confines the particles to a compact set away from the boundary. This excludes the known singularities that typically occur due to particles that repeatedly bounce off the boundary. In addition to the confinement, we follow the techniques introduced by Glassey and Schaeffer, who studied the Cauchy problem without boundaries.
Global classical solutions for the "One and one-half'' dimensional relativistic Vlasov-Maxwell-Fokker-Planck system
Stephen Pankavich and Nicholas Michalowski
2015, 8(1): 169-199 doi: 10.3934/krm.2015.8.169 +[Abstract](477) +[PDF](505.4KB)
Abstract:
In a recent paper Calogero and Alcántara [Kinet. Relat. Models, 4 (2011), pp. 401-426] derived a Lorentz-invariant Fokker-Planck equation, which corresponds to the evolution of a particle distribution associated with relativistic Brownian Motion. We study the ``one and one-half'' dimensional version of this problem with nonlinear electromagnetic interactions - the relativistic Vlasov-Maxwell-Fokker-Planck system - and obtain the first results concerning well-posedness of solutions. Specifically, we prove the global-in-time existence and uniqueness of classical solutions to the Cauchy problem and a gain in regularity of the distribution function in its momentum argument.

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