zbMATH — the first resource for mathematics

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

35F05 Linear first-order PDEs
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
82C70 Transport processes in time-dependent statistical mechanics
35B65 Smoothness and regularity of solutions to PDEs
Full Text: DOI
[1] Adams, R, Sobolev spaces, (1975), Academic Press New York/London · Zbl 0314.46030
[2] \scC. Bardos, F. Golse, B. Perthame, and R. Sentis, The nonaccretive radiative transfer equations. Existence of solutions and Rosseland approximation, to appear. · Zbl 0655.35075
[3] Bergh, J; Lofstrom, J, Interpolation spaces, (1976), Springer-Verlag New York/Berlin · Zbl 0344.46071
[4] Butzer, P.L; Berens, H, Semigroups of operators and approximation, (1967), 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, ()
[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 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.