On the global in time existence and uniqueness of solutions to the Boltzmann hierarchy

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Hauptverfasser: Ampatzoglou, Ioakeim, Miller, Joseph K., Pavlović, Nataša, Tasković, Maja
Format: Preprint
Veröffentlicht: 2024
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author Ampatzoglou, Ioakeim
Miller, Joseph K.
Pavlović, Nataša
Tasković, Maja
author_facet Ampatzoglou, Ioakeim
Miller, Joseph K.
Pavlović, Nataša
Tasković, Maja
contents In this paper we establish the global in time existence and uniqueness of solutions to the Boltzmann hierarchy, a hierarchy of equations instrumental for the rigorous derivation of the Boltzmann equation from many particles. Inspired by available $L^{\infty}$-based a-priori estimate for solutions to the Boltzmann equation, we develop the polynomially weighted $L^\infty$ a-priori bounds for solutions to the Boltzmann hierarchy and handle the factorial growth of the number of terms in the Dyson's series by reorganizing the sum through a combinatorial technique known as the Klainerman-Machedon board game argument. This paper is the first work that exploits such a combinatorial technique in conjunction with an $L^{\infty}$-based estimate to prove uniqueness of the mild solutions to the Boltzmann hierarchy. Our proof of existence of global in time mild solutions to the Boltzmann hierarchy for admissible initial data is constructive and it employs known global in time solutions to the Boltzmann equation via a Hewitt-Savage type theorem.
format Preprint
id arxiv_https___arxiv_org_abs_2402_00765
institution arXiv
publishDate 2024
record_format arxiv
spellingShingle On the global in time existence and uniqueness of solutions to the Boltzmann hierarchy
Ampatzoglou, Ioakeim
Miller, Joseph K.
Pavlović, Nataša
Tasković, Maja
Analysis of PDEs
In this paper we establish the global in time existence and uniqueness of solutions to the Boltzmann hierarchy, a hierarchy of equations instrumental for the rigorous derivation of the Boltzmann equation from many particles. Inspired by available $L^{\infty}$-based a-priori estimate for solutions to the Boltzmann equation, we develop the polynomially weighted $L^\infty$ a-priori bounds for solutions to the Boltzmann hierarchy and handle the factorial growth of the number of terms in the Dyson's series by reorganizing the sum through a combinatorial technique known as the Klainerman-Machedon board game argument. This paper is the first work that exploits such a combinatorial technique in conjunction with an $L^{\infty}$-based estimate to prove uniqueness of the mild solutions to the Boltzmann hierarchy. Our proof of existence of global in time mild solutions to the Boltzmann hierarchy for admissible initial data is constructive and it employs known global in time solutions to the Boltzmann equation via a Hewitt-Savage type theorem.
title On the global in time existence and uniqueness of solutions to the Boltzmann hierarchy
topic Analysis of PDEs
url https://arxiv.org/abs/2402.00765