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January  2009, 24(1): 115-143. doi: 10.3934/dcds.2009.24.115

Uniform $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with external forces

Seung-Yeal Ha 1, , Ho Lee 1, and  Seok Bae Yun 1,

1. 

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea, South Korea

Received  March 2007 Revised  September 2007 Published  January 2009

In this paper, we present an $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with cut-off and inverse power law potentials, when initial data are sufficiently small and decay fast enough in phase space. For moderately soft potentials, we show that classical solutions depend Lipschitz continuously on the initial data in weighted $L^p$-norm. In contrast for hard potentials, we show that classical solutions depend Hölder continuously on the initial data. Our stability estimates are based on the dispersion estimates due to time-asymptotic linear Vlasov dynamics.
Keywords: dilute gas., The Boltzmann equation, $L^p$-stability estimate.
Mathematics Subject Classification: Primary: 35Q72, 35Q80; Secondary: 35L4.
Citation: Seung-Yeal Ha, Ho Lee, Seok Bae Yun. Uniform $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with external forces. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 115-143. doi: 10.3934/dcds.2009.24.115
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