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

Pages: 353 - 364, Volume 11, Issue 2, March 2009

doi:10.3934/dcdsb.2009.11.353       Abstract        Full Text (202.3K)       Related Articles

Seung-Yeal Ha - Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea (email)
Mitsuru Yamazaki - Department of Mathematics and Computer Science, International Christian University, Tokyo 181-8585, Japan (email)

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.

Keywords:  The discrete velocity Boltzmann model, nonlinear functional, $L^p$ stability, macro-micro decomposition
Mathematics Subject Classification:  Primary: 76P05; Secondary: 37B25

Received: December 2007;      Revised: June 2008;      Published: December 2008.