Strong solutions of the Boltzmann equation in one spatial dimension

For the Boltzmann equation, the setting of a narrow shock tube implies that solutions $f(x,v,t)$ depend upon $v\in {\mathbb{R}}^3$, 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 $v\in {\mathbb{R}}^3$ and derivatives in both $x_1\in {\mathbb{R}}^1$ and $v\in {\mathbb{R}}^3$ 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. To cite this article: A. Biryuk et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).