Solutions to the non-cutoff Boltzmann equation uniformly near a Maxwellian

Luis Silvestre; Stanley Snelson; Luis Silvestre; Stanley Snelson

doi:10.3934/mine.2023034

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-36. doi: 10.3934/mine.2023034

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Solutions to the non-cutoff Boltzmann equation uniformly near a Maxwellian

Luis Silvestre ¹,
Stanley Snelson ^{2
,
,}

1.
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
2.
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

Received: 30 November 2021 Revised: 11 May 2022 Accepted: 11 May 2022 Published: 19 May 2022

The purpose of this paper is to show how the combination of the well-known results for convergence to equilibrium and conditional regularity, in addition to a short-time existence result, lead to a quick proof of the existence of global smooth solutions for the non cutoff Boltzmann equation when the initial data is close to equilibrium. We include a short-time existence result for polynomially-weighted $ L^\infty $ initial data. From this, we deduce that if the initial data is sufficiently close to a Maxwellian in this norm, then a smooth solution exists globally in time.
- Boltzmann equation,
- global existence,
- perturbation of equilibrium,
- conditional regularity,
- long-range interactions
Citation: Luis Silvestre, Stanley Snelson. Solutions to the non-cutoff Boltzmann equation uniformly near a Maxwellian[J]. Mathematics in Engineering, 2023, 5(2): 1-36. doi: 10.3934/mine.2023034

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Abstract

The purpose of this paper is to show how the combination of the well-known results for convergence to equilibrium and conditional regularity, in addition to a short-time existence result, lead to a quick proof of the existence of global smooth solutions for the non cutoff Boltzmann equation when the initial data is close to equilibrium. We include a short-time existence result for polynomially-weighted $ L^\infty $ initial data. From this, we deduce that if the initial data is sufficiently close to a Maxwellian in this norm, then a smooth solution exists globally in time.

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