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

We prove that if traceability conditions are fulfilled then a weak solution h is an element of L-infinity(R+ x R-d x R) to the ultra-parabolic transport equation partial derivative(t)h + div(x)(F(t, x, lambda)h) = Sigma(k)(i,j=1) partial derivative(2)(xixj) (b(ij)(t, x, lambda)h) + partial derivative(lambda)gamma(t, x, lambda), is such that for every rho is an element of C-c(1)(R), the velocity averaged quantity integral(R)h(t, x, lambda) rho(lambda)d lambda admits the strong L-loc(1) (R-d)-limit as t -> 0, i.e. there exist h(0)(x, lambda) is an element of L-loc(1)(R-d x R) and set E subset of R+ of full measure such that for every rho is an element of C-c(1)(R), L-loc(1)(R-d) - lim(t -> 0, t is an element of E)integral(R)h(t, x, lambda)rho(lambda)d lambda = integral(R)h(0)(x, lambda)rho(lambda)d lambda. As a corollary, under the traceability conditions, we prove the existence of strong traces for entropy solutions to ultra-parabolic equations in heterogeneous media.