On the diffusion approximation of a transport process without time scaling

Abstract

We are concerned with the asymptotic behaviour of the linear transport equation ∂_tu+ω▿u+(1/ε)Qu=0, u|_t=0=f, when ε→0. We prove that this process is equivalent up to order 1 to the viscosity perturbation of a hyperbolic process, provided that the conservative collision operator Q(x) has a 1-dimensional null-space. This can be considered as a justification of the diffusion approximation the classical scaling of the t or x variable.