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September  2009, 25(3): 869-882. doi: 10.3934/dcds.2009.25.869

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  January 2009 Revised  April 2009 Published  August 2009

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: 82A4.
Citation: Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869
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