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The main objective of this paper is to study the time decay oftransport-diffusion equation with inhomogeneous localized damping in themulti-dimensional torus. The drift is governed by an autonomous Lipschitzvector field and the diffusion by the standard heat equation with smallviscosity parameter $\nu$. In the first part we deal with the inviscid case andshow some results on the time decay of the energy using in a crucial way theergodicity and the unique ergodicity of the flow generated by the drift. In thesecond part we analyze the same problem with small viscosity and provide quitesimilar results on the exponential decay uniformly with respect to theviscosity in some logarithmic time scaling of the \mbox{type $t\in[0,C_0\ln(1/\nu)]$}.

Ergodicity effects on transport-diffusion equations with localized damping