STRONG TRACES FOR AVERAGED SOLUTIONS OF HETEROGENEOUS ULTRA-PARABOLIC TRANSPORT EQUATIONS

Abstract

We prove that if traceability conditions are fulfilled then a weak solution h ∈ L∞(ℝ+× ℝd× ℝ) to the ultra-parabolic transport equation [Formula: see text] is such that for every [Formula: see text], the velocity averaged quantity ∫ℝh(t, x, λ) ρ(λ)dλ admits the strong [Formula: see text]-limit as t → 0, i.e. there exist [Formula: see text] and set E ⊂ ℝ+of full measure such that for every [Formula: see text], [Formula: see text] As a corollary, under the traceability conditions, we prove the existence of strong traces for entropy solutions to ultra-parabolic equations in heterogeneous media.