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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Nonlinear wave propagations over a Boltzmann shock profile
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by Shih-Hsien Yu PDF
J. Amer. Math. Soc. 23 (2010), 1041-1118 Request permission

Abstract:

In this paper we study the wave propagation over a Boltzmann shock profile and obtain pointwise time-asymptotic stability of Boltzmann shocks. We design a ${\mathbb T}$-${\mathbb C}$ scheme to study the coupling of the transverse and compression waves. The pointwise information of the Green’s functions of the Boltzmann equation linearized around the end Maxwellian states of the shock wave provides the basic estimates for the transient waves. The compression of the Boltzmann shock profile together with a low order damping allows for an accurate energy estimate by a localized scalar equation. These two methods are combined to construct an exponentially sharp pointwise linear wave propagation structure around a Boltzmann shock profile. The pointwise estimates thus obtained are strong enough to study the pointwise nonlinear wave coupling and to conclude the convergence with an optimal convergent rate $O(1)[(1+t)(1+\varepsilon t)]^{-1/2}$ around the Boltzmann shock front, where $\varepsilon$ is the strength of a shock wave.
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Additional Information
  • Shih-Hsien Yu
  • Affiliation: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076
  • MR Author ID: 616205
  • Email: matysh@nus.edu.sg
  • Received by editor(s): February 24, 2009
  • Received by editor(s) in revised form: March 26, 2010
  • Published electronically: May 24, 2010
  • Additional Notes: This paper is supported by start-up grant R-146-000-108-133 of the National University of Singapore. The author thanks Professor Tai-Ping Liu for introducing him to the project on Boltzmann shock waves.
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 23 (2010), 1041-1118
  • MSC (2010): Primary 35L65, 35L67, 35Q20, 35E05
  • DOI: https://doi.org/10.1090/S0894-0347-2010-00671-6
  • MathSciNet review: 2669708