Compactness in Boltzmann's equation via Fourier integral operators and applications. III
type: Item
instance of: scholarly article
title: Compactness in Boltzmann's equation via Fourier integral operators and applications. III
zbMATH Open document ID: 0884.35124
author: P.-L. Lions
published in: Journal of Mathematics of Kyoto University
publication date: 1995-11-21T00:00:00Z
review text: [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.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76P05
Mathematics Subject Classification ID: 35S30
zbMATH DE Number: 746246
zbMATH Keywords: Vlasov-Boltzmann equations
zbMATH Keywords: Vlasov-Poisson-Boltzmann systems
zbMATH Keywords: existence
zbMATH Keywords: stability
zbMATH Keywords: Boltzmann-type equations
zbMATH Keywords: Vlasov-Maxwell-Boltzmann system
zbMATH Keywords: Boltzmann-Dirac equation
reviewed by: Reinhard Illner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518932
OpenAlex ID: W4251614073
DOI: 10.1215/KJM/1250518932
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description: scientific article; zbMATH DE number 746246