# Preprint 2008-027

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

## Darko Mitrović

**Abstract:**
We consider strong
L^{1}_{loc}(**R**^{d})
precompactness of the sequence of averaged quantities
∫_{Rm}h_{n}(x,λ)ρ(λ)dλ,
where ρ∈C_{0}(**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}^{ki} G^{i}_{n}(x,λ), x∈**R**^{d}, λ∈**R**^{m},

where F=(F_{1},…,F_{d}),
and F_{i}∈ L^{q}_{loc}(**R**^{d}×**R**^{m}), i=1,…,d,
1/p+1/q<1, and k_{i}=(k^{i}_{1},…,k^{i}_{m})∈**N**^{m},
i=1,…d, stands for multindex.
For the sequences of functions
(G_{n}^{i})_{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.