Partial Differential Equations
Strong solutions of the Boltzmann equation in one spatial dimension
Comptes Rendus. Mathématique, Volume 342 (2006) no. 11, pp. 843-848.

For the Boltzmann equation, the setting of a narrow shock tube implies that solutions $f(x,v,t)$ depend upon $v∈R3$, however they have one-dimensional spatial dependence. This Note discusses the case in which solutions are periodic in x, with controlled total energy and entropy, and such that the macroscopic density determined by the initial data is bounded. Our principal result is that the macroscopic density then remains bounded at all subsequent times, that is, this data gives rise to strong solutions which exist globally in time. Through a weak/strong uniqueness principle, these solutions are unique among the class of dissipative solutions. Additionally, we show that the flow of the Boltzmann equation propagates the moments in $v∈R3$ and derivatives in both $x1∈R1$ and $v∈R3$ of the solution $f(x,v,t)$. Our main theorems are valid for Boltzmann collision kernels which are bounded, and which have a relative velocity cutoff. The proofs depend upon a new averaging property of the collision operator and integral inequalities based in turn on entropy and on the Bony functional.

Dans un domaine qui représente un tube à choc, les solutions $f(x,v,t)$ de l'équation de Boltzmann dépendent de $v∈R3$ mais elles ne dépendent que de $x1∈R1$. Dans cette Note, on considère le cas de solutions périodiques en $x1∈R1$, dont la densité macroscopique initiale est finie, et l'énergie et l'entropie totales sont bornées par une certaine constante C. Le résultat principal est que la densité macroscopique de la solution reste bornée pour tout temps $t>0$, c'est-à-dire, les conditions initiales donnent lieu à des solutions fortes qui existent globalement en temps. Le résultat implique l'unicité de nos solutions dans la classe de solutions dissipatives faibles. Ces solutions $f(x,v,t)$ conservent les propriétés de régularité en x et en v, et les moments finis en v. Les théorèmes principaux sont valables pour des noyaux de collision de Boltzmann bornés, et avec une troncature de vitesse relative. Les démonstrations dépendent d'une propriété nouvelle de moyennisation de l'opérateur de collision, et de deux inégalités intégrales basées sur l'entropie et sur la fonctionnelle de Bony.

Accepted:
Published online:
DOI: 10.1016/j.crma.2006.04.005

Andrei Biryuk 1; Walter Craig 1; Vladislav Panferov 1

1 Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
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Andrei Biryuk; Walter Craig; Vladislav Panferov. Strong solutions of the Boltzmann equation in one spatial dimension. Comptes Rendus. Mathématique, Volume 342 (2006) no. 11, pp. 843-848. doi : 10.1016/j.crma.2006.04.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.04.005/

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