On a strong precompactness of velocity averages for a heterogenous transport equation with rough coefficients
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(Rd×Rm), p>1, are solutions to the transport equations with flux explicitly depending on space:
divx (F(x,λ)hn(x,λ))=∑i=1d ∂xi∂λki Gin(x,λ), x∈Rd, λ∈Rm,
where F=(F1,…,Fd), and Fi∈ Lqloc(Rd×Rm), 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(Rd×Rm) for a q~>1.
In order to obtain the result we adapt the notion of H-measures.