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Compactness in Boltzmann’s equation via Fourier integral operators and applications. III. (English) Zbl 0884.35124
[For parts I, II, cf. ibid., No. 2, 391-428, 429-462 (1994; Zbl 0831.35139).]
This third part deals with generalizations of existence and stability results for Boltzmann-type equations to
– Vlasov-Boltzmann (VB) equations, in particular Vlasov-Poisson-Boltzmann (VPB) systems
– the Vlasov-Maxwell-Boltzmann system (VMB) and
– the Boltzmann-Dirac equation (BD) (i.e., a Boltzmann equation with a correction corresponding to the Pauli exclusion principle).
The additional technical difficulties arising over the “pure” Boltzmann case are carefully discussed. Global existence of renormalized solutions is obtained for VB equations, but only a “very weak” global solution is shown to exist for VMB; specifically, a measure-valued solution satisfying physically meaningful estimates. For integrable collision kernels the existence of solutions for BD was established by Dolbeault (quoted in this article). Here, such a constraint on the collision kernel is dropped, and the compactness properties of the Boltzmann gain term proved in the author’s part one (loc. cit.) are adapted to obtain global solutions in the sense of distributions.

MSC:
35Q35 PDEs in connection with fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35S30 Fourier integral operators applied to PDEs
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