Kinetic equations and velocity averaging (Q2726213)

This noteworthy paper constitutes the author's doctoral thesis and deals, in great detail and with generalizations to the case of distribution spaces, with certain topics related to the transport equation \(v\cdot\nabla_x f= g\) and the ``velocity averaging'' problem. The preceding equation, more or less in its classical form, is replaced by a more general transport equation, which has the form NEWLINE\[NEWLINE\text{div}_x (af)=g,\tag{1}NEWLINE\]NEWLINE in \(S'(\mathbb{R}^n,E)\), where \(E=E (\mathbb{R})\), is a rearrangement (in a precise sense) of a measure space \((\mathbb{R}, \mu)\), where \(\mu\) is nonatomic and finite. \(S'(\mathbb{R}^n,E)\) stands for the Schwartz space of distributions.NEWLINENEWLINENEWLINEThe meaning of (1) is clarified by introducing and investigating various new concepts. In (1), \(a\) stands for a map from \(\mathbb{R}\) into \(\mathbb{R}^n\). Among the concepts considered is that of a mean value, crucial in dealing with ``velocity averaging''. Further basic concepts are: Banach-space valued tempered distributions; Besov spaces of Banach-space valued functions (only this item requires no less than 10 pages); regularity for ``velocity averaging''. The reading of the paper requires serious knowledge of functional analysis, measure theory and other related topics. The new concepts, introduced by the author, are clearly defined and thorougly investigated. The paper represents a solid contribution to both transport theory and functional analysis.