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Preprint 2008-027

On a strong precompactness of velocity averages for a heterogenous transport equation with rough coefficients

Darko Mitrović

Abstract: We consider strong L1loc(Rd) precompactness of the sequence of averaged quantities Rmhn(x,λ)ρ(λ)dλ, where ρ∈C0(Rm), and hn∈Lploc(RdRm), p>1, are solutions to the transport equations with flux explicitly depending on space:

divx (F(x,λ)hn(x,λ))=∑i=1dxiλki Gin(x,λ),x∈Rd,λ∈Rm,

where F=(F1,…,Fd), and Fi∈ Lqloc(RdRm), i=1,…,d, 1/p+1/q<1, and ki=(ki1,…,kim)∈Nm, i=1,…d, stands for multindex. For the sequences of functions (Gni)n∈N, i=1,…,d, we assume that they strongly converge to zero as n→∞ in Lq~loc(RdRm) for a q~>1.

In order to obtain the result we adapt the notion of H-measures.

Paper:
Available as PDF (310 Kbytes).
Author(s):
Darko Mitrović
Publishing information:
Comments:
Updated 2009-01-08.
Submitted by:
; 2008-09-10.