Analysis of the advection-diffusion operator using fractional order norms

Summary. In this paper we obtain a family of optimal estimates for the linear advection-diffusion operator. More precisely we define norms on the domain of the operator, and norms on its image, such that it behaves as an isomorphism: it stays bounded as well as its inverse does, uniformly with respect to the diffusion parameter. The analysis makes use of the interpolation theory between function spaces. One motivation of the present work is our interest in the theoretical properties of stable numerical methods for this kind of problem: we will only give some hints here and we will take a deeper look in a further paper.Mathematics Subject Classification (2000):65N30