Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions

Marc Briant

doi:10.3934/krm.2015.8.281

Article Contents

2015, Volume 8, Issue 2: 281-308. Doi: 10.3934/krm.2015.8.281

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Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions

Marc Briant^1,

1.
Division of Applied Mathematics, Brown University, Box F, 182 George Street, Providence, RI 02912

Received: September 30, 2014

Revised: December 31, 2014

Published: May 2015

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Abstract

We prove the immediate appearance of an exponential lower bound, uniform in time and space, for continuous mild solutions to the full Boltzmann equation in a $C^2$ convex bounded domain with the physical Maxwellian diffusion boundary conditions, under the sole assumption of regularity of the solution. We investigate a wide range of collision kernels, with and without Grad's angular cutoff assumption. In particular, the lower bound is proven to be Maxwellian in the case of cutoff collision kernels. Moreover, these results are entirely constructive if the initial distribution contains no vacuum, with explicit constants depending only on the a priori bounds on the solution.

Keywords:

Boltzmann equation,
exponential lower bound,
Maxwellian lower bound,
Maxwellian diffusion boundary conditions.

Mathematics Subject Classification: Primary: 35B09, 35Q20; Secondary: 82B40.

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References

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