Volume 1, Issue 1
On Smooth Solutions to the Thermostated Boltzmann Equation with Deformation

Renjun Duan & Shuangqian Liu

Commun. Math. Anal. Appl., 1 (2022), pp. 152-212.

Published online: 2022-01

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  • Abstract

This paper concerns a kinetic model of the thermostated Boltzmann equation with a linear deformation force described by a constant matrix. The collision kernel under consideration includes both the Maxwell molecule and general hard potentials with angular cutoff. We construct the smooth steady solutions via a perturbation approach when the deformation strength is sufficiently small. The steady solution is a spatially homogeneous non Maxwellian state and may have the polynomial tail at large velocities. Moreover, we also establish the long time asymptotics toward steady states for the Cauchy problem on the corresponding spatially inhomogeneous equation in torus, which in turn gives the non-negativity of steady solutions.

  • AMS Subject Headings

35Q20, 35B40

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COPYRIGHT: © Global Science Press

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@Article{CMAA-1-152, author = {Duan , Renjun and Liu , Shuangqian}, title = {On Smooth Solutions to the Thermostated Boltzmann Equation with Deformation}, journal = {Communications in Mathematical Analysis and Applications}, year = {2022}, volume = {1}, number = {1}, pages = {152--212}, abstract = {

This paper concerns a kinetic model of the thermostated Boltzmann equation with a linear deformation force described by a constant matrix. The collision kernel under consideration includes both the Maxwell molecule and general hard potentials with angular cutoff. We construct the smooth steady solutions via a perturbation approach when the deformation strength is sufficiently small. The steady solution is a spatially homogeneous non Maxwellian state and may have the polynomial tail at large velocities. Moreover, we also establish the long time asymptotics toward steady states for the Cauchy problem on the corresponding spatially inhomogeneous equation in torus, which in turn gives the non-negativity of steady solutions.

}, issn = {2790-1939}, doi = {https://doi.org/10.4208/cmaa.2021-0004}, url = {http://global-sci.org/intro/article_detail/cmaa/20160.html} }
TY - JOUR T1 - On Smooth Solutions to the Thermostated Boltzmann Equation with Deformation AU - Duan , Renjun AU - Liu , Shuangqian JO - Communications in Mathematical Analysis and Applications VL - 1 SP - 152 EP - 212 PY - 2022 DA - 2022/01 SN - 1 DO - http://doi.org/10.4208/cmaa.2021-0004 UR - https://global-sci.org/intro/article_detail/cmaa/20160.html KW - Boltzmann equation, deformation force, thermostated force, non-equilibrium steady state, asymptotic stability. AB -

This paper concerns a kinetic model of the thermostated Boltzmann equation with a linear deformation force described by a constant matrix. The collision kernel under consideration includes both the Maxwell molecule and general hard potentials with angular cutoff. We construct the smooth steady solutions via a perturbation approach when the deformation strength is sufficiently small. The steady solution is a spatially homogeneous non Maxwellian state and may have the polynomial tail at large velocities. Moreover, we also establish the long time asymptotics toward steady states for the Cauchy problem on the corresponding spatially inhomogeneous equation in torus, which in turn gives the non-negativity of steady solutions.

Renjun Duan & Shuangqian Liu. (2022). On Smooth Solutions to the Thermostated Boltzmann Equation with Deformation. Communications in Mathematical Analysis and Applications. 1 (1). 152-212. doi:10.4208/cmaa.2021-0004
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