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Nonlinear wave propagations over a Boltzmann shock profile

Author: Shih-Hsien Yu
Journal: J. Amer. Math. Soc. 23 (2010), 1041-1118
MSC (2010): Primary 35L65, 35L67, 35Q20, 35E05
Published electronically: May 24, 2010
MathSciNet review: 2669708
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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

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.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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