Relaxation in Sobolev spaces and $ L^1 $ spectral gap of the 1D dissipative Boltzmann equation with Maxwell interactions

R. Alonso; V. Bagland; J. A. Cañizo; B. Lods; S. Throm

doi:10.3934/dcds.2025068

Article Contents

2025, Volume 45, Issue 11: 4572-4605. Doi: 10.3934/dcds.2025068

This issue Previous Article Nonexistence results for a nonvariational system with variable exponents and existence via blow-up Next Article

Relaxation in Sobolev spaces and $ L^1 $ spectral gap of the 1D dissipative Boltzmann equation with Maxwell interactions

1.
Division of Arts & Sciences, Texas A & M University at Qatar, Education City, Doha, Qatar
2.
Université Clermont Auvergne, LMBP, UMR 6620 - CNRS, Campus des Cézeaux, 3, place Vasarely, TSA 60026, CS 60026, F-63178 Aubière Cedex, France
3.
Departamento de Matemática Aplicada & IMAG, Universidad de Granada, Avenida de Fuentenueva S/N, 18071 Granada, Spain
4.
Università degli Studi di Torino & Collegio Carlo Alberto, Department of Economics, Social Sciences, Applied Mathematics and Statistics "ESOMAS", Corso Unione Sovietica, 218/bis, 10134 Torino, Italy
5.
Umeå University, Department of Mathematics and Mathematical Statistics, 901 87 Umeå, Sweden

^*Corresponding author: Bertrand Lods
^*Corresponding author: Bertrand Lods

Received: April 2024

Revised: March 2025

Early access: May 2025

Published: November 2025

R. Alonso gratefully acknowledges the support from O Conselho Nacional de Desenvolvimento Científico e Tecnológico, Bolsa de Produtividade em Pesquisa - CNPq (303325/2019-4). J. Cañizo acknowledges support from grant PID2020-117846GB-I00, the research network RED2018-102650-T, and the María de Maeztu grant CEX2020-001105-M from the Spanish government. B. Lods was partially supported by PRIN2022 (project ID: BEMMLZ) Stochastic control and games and the role of information. He also gratefully acknowledges the financial support from the Italian Ministry of Education, University and Research (MIUR), Dipartimenti di Eccellenza grant 2022-2027 as well as the support from the de Castro Statistics Initiative, Collegio Carlo Alberto (Torino). The authors would like to acknowledge the support of the Hausdorff Institute for Mathematics where this work started during their stay at the 2019 Junior Trimester Program on Kinetic Theory. The authors wish to thank the reviewers for the careful reading and suggestions which helped improve the presentation of some of the results.

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Abstract

We study the dynamic relaxation to equilibrium of the 1D dissipative Boltzmann equation with Maxwell interactions in classical $ H^s $ Sobolev spaces. In addition, we present a shrinkage analysis and decay estimates for the linearised 1D dissipative Boltzmann operator with such interactions. Based on this study, we explore the convergence in $ H^s $ and $ L^{1} $ spaces for the linear and nonlinear models. This study extends classical results found in the literature given for spaces with weak topologies.

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Mathematics Subject Classification: 35Q20, 82C22, 82C40, 45G05, 40H05.

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References

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