# American Institute of Mathematical Sciences

ISSN:
1937-5093

eISSN:
1937-5077

All Issues

## Kinetic & Related Models

December 2010 , Volume 3 , Issue 4

Special issue
on KAM theory and its applications

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2010, 3(4): 529-644 doi: 10.3934/krm.2010.3.529 +[Abstract](1714) +[PDF](1175.6KB)
Abstract:
N/A
2010, 3(4): 645-667 doi: 10.3934/krm.2010.3.645 +[Abstract](1484) +[PDF](403.9KB)
Abstract:
In this paper, we consider the regularity of solutions to the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. In particular, we get the analytic smoothing effects for solutions obtained by Bagland if we assume all the moments for the initial datum are finite.
2010, 3(4): 669-683 doi: 10.3934/krm.2010.3.669 +[Abstract](1297) +[PDF](407.8KB)
Abstract:
The purpose of this note is to extend the $L^1$ averaging lemma of Golse and Saint-Raymond [10] to the case of a kinetic transport equation with a force field $F(x)\in W^{1,\infty}$. To this end, we will prove a local in time mixing property for the transport equation $\partial_t f + v.\nabla_x f + F.\nabla_v f =0$.
2010, 3(4): 685-728 doi: 10.3934/krm.2010.3.685 +[Abstract](2149) +[PDF](606.8KB)
Abstract:
Fluid dynamic limit to compressible Euler equations from compressible Navier-Stokes equations and Boltzmann equation has been an active topic with limited success so far. In this paper, we consider the case when the solution of the Euler equations is a Riemann solution consisting two rarefaction waves and a contact discontinuity and prove this limit for both Navier-Stokes equations and the Boltzmann equation when the viscosity, heat conductivity coefficients and the Knudsen number tend to zero respectively. In addition, the uniform convergence rates in terms of the above physical parameters are also obtained. It is noted that this is the first rigorous proof of this limit for a Riemann solution with superposition of three waves even though the fluid dynamic limit for a single wave has been proved.
2010, 3(4): 729-754 doi: 10.3934/krm.2010.3.729 +[Abstract](1425) +[PDF](499.7KB)
Abstract:
We construct global weak solutions for both one-component and two-component Vlasov-Poisson equations in a single space dimension with electron sheet initial data. We give an explicit formula of the weak solution of the one-component Vlasov-Poisson equation provided the electron sheet remains a graph in the $x$-$v$ plane, and we give sharp conditions on whether the moment of this explicit weak solution will blow up or not. We introduce new parameters, which we call "charge indexes", to construct the global weak solution. The moment of the weak solution corresponds to a multi-valued solution to the Euler-Poisson system. Our method guarantees that even if concentration in charge develops, it will disappear immediately. We extend our method to more singular initial data, where charge can concentrate on points at time $t=0$. Examples show that for one-component Vlasov-Poisson equation our weak solution agrees with the continuous fission weak solution, which is the zero diffusion limit of the Fokker-Planck equation. Finally, we propose a novel numerical method to compute solutions of both one-component and two-component Vlasov-Poisson equations and the multi-valued solution of the one-dimensional Euler-Poisson equation.

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