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Kinetic & Related Models

December 2009 , Volume 2 , Issue 4

A special issue on Bifurcation Delay

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A review of Boltzmann equation with singular kernels
Radjesvarane Alexandre
2009, 2(4): 551-646 doi: 10.3934/krm.2009.2.551 +[Abstract](751) +[PDF](940.2KB)
Abstract:
We review recent results about Boltzmann equation for singular or non cutoff cross-sections. Both spatially homogeneous and inhomogeneous Boltzmann equations are considered, and ideas related to Landau equation are explained. Various technical tools are presented, together with applications to existence and regularization issues.
Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation
Nadia Lekrine and Chao-Jiang Xu
2009, 2(4): 647-666 doi: 10.3934/krm.2009.2.647 +[Abstract](587) +[PDF](246.5KB)
Abstract:
In this work, we consider a spatially homogeneous Kac's equation with a non cutoff cross section. We prove that the weak solution of the Cauchy problem is in the Gevrey class for positive time. This is a Gevrey regularizing effect for non smooth initial datum. The proof relies on the Fourier analysis of Kac's operators and on an exponential type mollifier.
Existence and sharp localization in velocity of small-amplitude Boltzmann shocks
Guy Métivier and K. Zumbrun
2009, 2(4): 667-705 doi: 10.3934/krm.2009.2.667 +[Abstract](572) +[PDF](465.9KB)
Abstract:
Using a weighted $H^s$-contraction mapping argument based on the macro-micro decomposition of Liu and Yu, we give an elementary proof of existence, with sharp rates of decay and distance from the Chapman-Enskog approximation, of small-amplitude shock profiles of the Boltzmann equation with hard-sphere potential, recovering and slightly sharpening results obtained by Caflisch and Nicolaenko using different techniques. A key technical point in both analyses is that the linearized collision operator $L$ is negative definite on its range, not only in the standard square-root Maxwellian weighted norm for which it is self-adjoint, but also in norms with nearby weights. Exploring this issue further, we show that $L$ is negative definite on its range in a much wider class of norms including norms with weights asymptotic nearly to a full Maxwellian rather than its square root. This yields sharp localization in velocity at near-Maxwellian rate, rather than the square-root rate obtained in previous analyses.
Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution
Ghendrih Philippe, Hauray Maxime and Anne Nouri
2009, 2(4): 707-725 doi: 10.3934/krm.2009.2.707 +[Abstract](652) +[PDF](249.8KB)
Abstract:
A finite Larmor radius approximation is rigourously derived from the Vlasov equation, in the limit of large (and uniform) external magnetic field. Existence and uniqueness of a solution is proven in the stationary frame.

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