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Steady solutions of the Boltzmann equation for a gas flow past an obstacle. I: Existence. (English) Zbl 0538.76070
This paper presents the solution f of the nonlinar Boltzmann equation for stationary flow past a convex obstacle. On the obstacle various boundary conditions are considered, such as specular reflection or diffusion reflection. At infinity the distribution is assumed to be Maxwellian, with a small velocity relative to the obstacle but with the same temperature as the obstacle (for the diffuse reflection case). In the proof the equation is perturbed by addition of a term \(\lambda\) f. After this perturbed problem is solved, the limit \(\lambda\to 0\) is taken. This may be thought of as introducing time variation and letting \(t\to \infty\). The perturbed problem is solved by splitting the problem into two simpler pieces using resolvent identities. The two pieces are the full linearized equation with no obstacle and the linearized equation with the obstacle but with only the ”absorption” part of the collision integral.
This is a difficult and important piece of work. It stimulates many questions on qualitative properties of the solution and the solution of other boundary value problems.
Reviewer: R.Caflish

MSC:
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
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