Acoustic limit of the Boltzmann equation: Classical solutions

Juhi Jang; Ning Jiang

doi:10.3934/dcds.2009.25.869

Article Contents

2009, Volume 25, Issue 3: 869-882. Doi: 10.3934/dcds.2009.25.869

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Acoustic limit of the Boltzmann equation: Classical solutions

Juhi Jang^1, and
Ning Jiang^1,

1.
Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York City, NY 10012

Received: December 31, 2008

Revised: March 31, 2009

Published: August 2009

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Abstract

We study the acoustic limit from the Boltzmann equation in the framework of classical solutions. For a solution $F_\varepsilon=\mu +\varepsilon \sqrt{\mu}f_\varepsilon$ to the rescaled Boltzmann equation in the acoustic time scaling

$\partial_t F_\varepsilon +\v$•$grad$_x$F_\varepsilon =\frac{1}{\varepsilon} \Q(F_\varepsilon,F_\varepsilon)\,$

inside a periodic box $\mathbb{T}^3$, we establish the global-in-time uniform energy estimates of $f_\varepsilon$ in $\varepsilon$ and prove that $f_\varepsilon$ converges strongly to $f$ whose dynamics is governed by the acoustic system. The collision kernel $\Q$ includes hard-sphere interaction and inverse-power law with an angular cutoff.

Keywords:

fluid dynamic limit,
Boltzmann equations.

Mathematics Subject Classification: Primary: 76P05, 35L67; Secondary: 82A40.

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