Abstract: We consider strong L¹_loc(R^d) precompactness of the sequence of averaged quantities ∫_R^mh_n(x,λ)ρ(λ)dλ, where ρ∈C₀(R^m), and h_n∈L^p_loc(R^d×R^m), p>1, are solutions to the transport equations with flux explicitly depending on space:

div_x (F(x,λ)h_n(x,λ))=∑_i=1^d ∂_xi∂_λ^k_i Gⁱ_n(x,λ),    x∈R^d,    λ∈R^m,

where F=(F₁,…,F_d), and F_i∈ L^q_loc(R^d×R^m), i=1,…,d, 1/p+1/q<1, and k_i=(kⁱ₁,…,kⁱ_m)∈N^m, i=1,…d, stands for multindex. For the sequences of functions (G_nⁱ)_n∈N, i=1,…,d, we assume that they strongly converge to zero as n→∞ in L^q~_loc(R^d×R^m) for a q~>1.

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