METIVIER, Guy; ZUMBRUN, Kevin

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METIVIER, Guy
Institut de Mathématiques de Bordeaux [IMB]

ZUMBRUN, Kevin
Department of Mathematics IU

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Kinetic and Related Models. 2009, vol. 2, p. pp 209--231

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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.< Réduire

Mots clés en anglais

shock profiles

relaxation

macro-micro decomposition

Chapman-Enskog approximation

energy methods

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Existence and sharp localization in velocity of small amplitude Boltzman shocks

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