Kinetic and Related Models (KRM)

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

Pages: 667 - 705, Volume 2, Issue 4, December 2009      doi:10.3934/krm.2009.2.667

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Guy Métivier - IMB, Université de Bordeaux, CNRS, IMB, 33405 Talence Cedex, France (email)
K. Zumbrun - Mathematics Department, Indiana University, Bloomington, IN 47405, United States (email)

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.

Keywords:  Boltzmann equation, shock waves, Chapman--Enskog approximation.
Mathematics Subject Classification:  Primary: 82C40; Secondary: 35Q99.

Received: July 2009;      Revised: August 2009;      Available Online: October 2009.