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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
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