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The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball. III: The 3-D Boltzmann equation. (English) Zbl 1379.35259

The purpose of this article is to continue the studies of compressible fluids in an infinite ball using the Boltzmann equation. After a change of variables the Boltzmann equation becomes (1). The main theorem proves the existence of a unique, global mild solution of (1) with certain initial boundary data. The proofs consist of several steps and use backward trajectory, null-space of linearized collision operator, \(L^2\)-estimate of solutions to the linear Boltzmann equation, and conservation laws of mass and energy.
For Part I and II see [the first author and G. Xu, “The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball. I: 3D Euler equations”, Preprint, arXiv:1706.01183; the first author and {Ł. Zhang}, “The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball. II: 3D Navier-Stokes equations”, Discrete Contin. Dyn. Syst. 28, No. 3, 1063–1102 (2018; doi:10.3934/dcds.2018045)].

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q20 Boltzmann equations
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35L70 Second-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
76N15 Gas dynamics (general theory)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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