×

Regularity of the moments of the solution of a transport equation. (English) Zbl 0652.47031

Author’s abstract: Let \(u=u(x,v)\) satisfy the transport equation \(u+v\cdot \partial_ xu=f\), \(x\in {\mathbb R}^ N\), \(r\in {\mathbb R}^ N\), where \(f\) belongs to some space of type \(L^ p(dx\otimes d\mu (v))\) (where \(\mu\) is a positive bounded measure on \({\mathbb R}^ N)\). We study the resulting regularity of the moment \(\int u(x,v)\,d\mu (v)\) (in terms of fractional Sobolev spaces, for example). Counterexamples are given in order to test the optimality of our results.
Reviewer: R.Weikard

MSC:

35F05 Linear first-order PDEs
47F05 General theory of partial differential operators
82C70 Transport processes in time-dependent statistical mechanics
35B65 Smoothness and regularity of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, R., Sobolev Spaces (1975), Academic Press: Academic Press New York/London · Zbl 0314.46030
[3] Bergh, J.; Lofstrom, J., Interpolation Spaces (1976), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0344.46071
[4] Butzer, P. L.; Berens, H., Semigroups of Operators and Approximation (1967), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0164.43702
[5] Cessenat, M., Théorème de traces pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris, 300, 1, 89-92 (1985) · Zbl 0648.46028
[6] Dautray, R.; Lions, J. L., (Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Vol. III (1985), Masson: Masson Paris) · Zbl 0642.35001
[7] Golse, F.; Perthame, B.; Sentis, R., Un résultat de compacité pour les équations de transport et application au calcul de la valeur propre principale d’un opérateur de transport, C. R. Acad. Sci. Paris, 301, 341-344 (1985) · Zbl 0591.45007
[8] Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (1978), North-Holland: North-Holland Amsterdam · Zbl 0387.46032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.