# American Institute of Mathematical Sciences

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