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July  2006, 14(3): 579-596. doi: 10.3934/dcds.2006.14.579

Time-periodic solutions of the Boltzmann equation

Seiji Ukai 1,

1. 

Liu Bie Ju Centre for Mathematical Sciences and Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China

Received  November 2004 Revised  March 2005 Published  December 2005

The Boltzmann equation with a time-periodic inhomogeneous term is solved on the existence of a time-periodic solution that is close to an absolute Maxwellian and has the same period as the inhomogeneous term, under some smallness assumption on the inhomogeneous term and for the spatial dimension $n\ge 5$, and also for the case $n=3$ and $ 4$ with an additional assumption that the spatial integral of the macroscopic component of the inhomogeneous term vanishes. This solution is a unique time-periodic solution near the relevant Maxwellian and asymptotically stable in time. Similar results are established also with the space-periodic boundary condition. As a special case, our results cover the case where the inhomogeneous term is time-independent, proving the unique existence and asymptotic stability of stationary solutions. The proof is based on a combination of the contraction mapping principle and time-decay estimates of solutions to the linearized Boltzmann equation.
Keywords: time-periodic solution, asymptotic stability, time-decay estimate., stationary solution, Boltzmann equation.
Mathematics Subject Classification: 35Q35, 76P05, 82C4.
Citation: Seiji Ukai. Time-periodic solutions of the Boltzmann equation. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 579-596. doi: 10.3934/dcds.2006.14.579
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