Dynamics of Partial Differential Equations

Volume 18 (2021)

Number 1

Ergodicity effects on transport-diffusion equations with localized damping

Pages: 1 – 10

DOI: https://dx.doi.org/10.4310/DPDE.2021.v18.n1.a1


Kaïs Ammari (UR Analysis and Control of PDEs, Department of Mathematics, Faculty of Sciences, University of Monastir, Tunisia)

Taoufik Hmidi (Institut de recherche mathématique de Rennes (IRMAR), Université de Rennes, France)


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


ergodicity effects, transport-diffusion equations, localized damping

2010 Mathematics Subject Classification

35B40, 37A10, 37A30

Received 28 October 2020

Published 19 February 2021