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Conditions at infinity for Boltzmann’s equation. (English) Zbl 0799.35210
The paper studies boundary conditions at infinity for the Boltzmann equation: ${{\partial f} \over {\partial t}}+ v\cdot \nabla_ x f= Q(f,f), \qquad x\in \mathbb{R}^ N, \quad v\in \mathbb{R}^ N, \quad t\in (0,\infty), \tag{B}$ where $$N\geq 1$$ and $$Q(f,f)$$ is the so-called collision operator. An initial condition is given: (1) $$f|_{t=0}= f_ 0(x,\xi)$$ in $$\mathbb{R}^ N\times \mathbb{R}^ N$$. The following model of behaviour at infinity is studied: (2) $$f\to M$$ as $$| x|\to +\infty$$, where $$M$$ is a pure Maxwellian. This problem is called the case of a Maxwellian at infinity. Another problem consists of looking at (B), (1) in $$\Omega\times \mathbb{R}_ v^ N\times (0,\infty)$$, ($$\Omega={\mathcal O}^ c$$, where $${\mathcal O}$$ a bounded smooth open set of $$\mathbb{R}^ N$$), with (2) and a boundary condition on $$\partial\Omega\times \mathbb{R}_ v^ N\times (0,\infty)$$ like a specular reflection condition. This is called the exterior domain case.
Moreover, the “tube case” is studied, that is (B), (1) in $$\Omega= \mathbb{R}\times \omega$$ where $$\omega$$ is a bounded smooth open set in $$\mathbb{R}^{N-1}$$ with specular reflection condition in $$\partial\Omega= \mathbb{R}\times \partial\omega$$ and (2) replaced by (3) $$f\to M^ +$$ as $$x_ 1\to +\infty$$, $$f\to M^ -$$ as $$x_ 1\to +\infty$$, where $$M^ +$$, $$M^ -$$ are two pure Maxwellians of the following form $M^ \pm= \rho^ \pm \exp (| v_ 1- u_ 1^ \pm|^ 2+ | v_ 2|^ 2+ \dots+ | v_ N|^ 2/ 2T^ \pm) (2\pi T^ \pm)^{-N/2}\tag{4}$ with $$\rho^ +,\rho^ ->0$$, $$T^ +, T^ ->0$$, $$u_ 1^ +, u_ 1^ -\in \mathbb{R}$$.
Reviewer: S.Totaro (Firenze)

##### MSC:
 35Q72 Other PDE from mechanics (MSC2000) 35B40 Asymptotic behavior of solutions to PDEs 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 82C70 Transport processes in time-dependent statistical mechanics
##### Keywords:
Boltzmann equation; boundary conditions at infinity
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