Velocity averaging -- a general framework

2011/07/13 by Martin Lazar, Darko Mitrovic, Lazar, Martin +1 · 2 citations
Mathematics · #35A22 #35A27 #35K70 #42B37 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #math.AP #math.FA #msc:35A22 #msc:35A27 #msc:35K70 #msc:42B37

paper · pdf · doi:10.48550/arxiv.1107.2616

generality is decreased and mistakes are corrected; to appear in Dyn of PDE

arxiv created 2012/09/22 · arxiv updated 2012/09/25

Abstract

We prove that the sequence of averaged quantities ∫_\R^mu_n(\mx,\msnop) ρ(\msnop)d\msnop, is strongly precompact in \Ldl\Rd, where ρ∈ \Ldc\R^m, and u_n∈ \Ld\R^m; \pL s\Rd, s≥ 2, are weak solutions to differential operator equations with variable coefficients. In particular, this includes differential operators of hyperbolic, parabolic or ultraparabolic type, but also fractional differential operators. If s>2 then the coefficients can be discontinuous with respect to the space variable \mx∈ \R^d, otherwise, the coefficients are continuous functions. In order to obtain the result we prove a representation theorem for an extension of the H-measures.

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