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

Seung-Yeal Ha; Ho Lee; Seok Bae Yun

doi:10.3934/dcds.2009.24.115

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2009, Volume 24, Issue 1: 115-143. Doi: 10.3934/dcds.2009.24.115

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

1.
Department of Mathematical Sciences, Seoul National University, Seoul 151-747

Received: February 28, 2007

Revised: August 31, 2007

Published: December 2008

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q72, 35Q80; Secondary: 35L45.

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