$L^p$-stability estimates for the spatially inhomogeneous discretevelocity Boltzmann model

Seung-Yeal Ha; Mitsuru Yamazaki

doi:10.3934/dcdsb.2009.11.353

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2009, Volume 11, Issue 2: 353-364. Doi: 10.3934/dcdsb.2009.11.353

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$L^p$-stability estimates for the spatially inhomogeneous discrete velocity Boltzmann model

Seung-Yeal Ha^1, and
Mitsuru Yamazaki^2,

1.
Department of Mathematical Sciences, Seoul National University, Seoul 151-747
2.
Department of Mathematics and Computer Science, International Christian University, Tokyo 181-8585

Received: December 2007

Revised: June 2008

Published: September 2009

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Abstract

We present two a priori $L^p$-stability estimates to the discrete velocity Boltzmann models. In a close-to-global Maxwellian regime, we derive a local-in-time $L^2$-stability estimate using a macro-micro decomposition and dispersion estimates for smooth perturbations, and as a direct application, we establish that classical solutions in Kawashima's framework [22, 24] are uniformly $L^2$-stable. In a close-to-vacuum regime, we also obtain a local-in-time $L^p$-stability estimates for classical solutions near vacuum.

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Mathematics Subject Classification: Primary: 76P05; Secondary: 37B25.

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