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Kinetic & Related Models

2010 , Volume 3 , Issue 4

Special issue
on KAM theory and its applications

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Kinetic limits for waves in a random medium
Guillaume Bal , Tomasz Komorowski and  Lenya Ryzhik
2010, 3(4): 529-644 doi: 10.3934/krm.2010.3.529 +[Abstract](119) +[PDF](1175.6KB)
Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials
Yemin Chen
2010, 3(4): 645-667 doi: 10.3934/krm.2010.3.645 +[Abstract](28) +[PDF](403.9KB)
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.
$L^1$ averaging lemma for transport equations with Lipschitz force fields
Daniel Han-Kwan
2010, 3(4): 669-683 doi: 10.3934/krm.2010.3.669 +[Abstract](37) +[PDF](407.8KB)
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 $\partial_t f + v.\nabla_x f + F.\nabla_v f =0$.
Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity
Feimin Huang , Yi Wang and  Tong Yang
2010, 3(4): 685-728 doi: 10.3934/krm.2010.3.685 +[Abstract](77) +[PDF](606.8KB)
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.
1D Vlasov-Poisson equations with electron sheet initial data
Dongming Wei
2010, 3(4): 729-754 doi: 10.3934/krm.2010.3.729 +[Abstract](38) +[PDF](499.7KB)
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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