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December  2009, 2(4): 667-705. doi: 10.3934/krm.2009.2.667

Existence and sharp localization in velocity of small-amplitude Boltzmann shocks

1. 

IMB, Université de Bordeaux, CNRS, IMB, 33405 Talence Cedex, France

2. 

Mathematics Department, Indiana University, Bloomington, IN 47405

Received  July 2009 Revised  August 2009 Published  October 2009

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.
Citation: Guy Métivier, K. Zumbrun. Existence and sharp localization in velocity of small-amplitude Boltzmann shocks. Kinetic and Related Models, 2009, 2 (4) : 667-705. doi: 10.3934/krm.2009.2.667
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