zbMATH — the first resource for mathematics

On the trace problem for solutions of the Vlasov equation. (English) Zbl 0953.35028
Summary: We study the trace problem for weak solutions of the Vlasov equation set in a domain. When the force field has Sobolev regularity, we prove the existence of a trace on the boundaries, which is defined thanks to a Green formula, and we show that the trace can be renormalized. We apply these results to prove existence and uniqueness of the Cauchy problem for the Vlasov equation with specular reflection at the boundary. We also give optimal trace theorems and solve the Cauchy problem with general Dirichlet conditions at the boundary.

35D10 Regularity of generalized solutions of PDE (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
82D10 Statistical mechanics of plasmas
Full Text: DOI
[1] V.I. Agoshkov, Functional spaces H 1 p ,Hp t+{\(\alpha\)},k and resolution conditions of boundary problems for transport equation, prepublication du Department of Numerical Mathematics, USSR Academy of Sciences, Moscow.
[2] DOI: 10.1007/BF00383222 · Zbl 0789.76075
[3] DOI: 10.1090/S0002-9947-1982-0645322-8
[4] Brézis H, prepublication of the Laboratoire dlAnalyse Numkrique, UniversitC P. et M. Curie, 12 (1993)
[5] DOI: 10.1016/0893-9659(91)90077-9 · Zbl 0744.45005
[6] Cessenat M, Note C.R.Acad.Sci.Paris 300 pp 89– (1985)
[7] DOI: 10.1007/BF01393835 · Zbl 0696.34049
[8] DOI: 10.1007/BF01837113 · Zbl 0777.76084
[9] Heintz A, Ph.D.Thesis, Leningrad State University, (1986)
[10] Lions P.L, dispersion, Note C. R. Acad. Sci. Paris 314 pp 801– (1992)
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.