Diusion and mixing in fluid flow

Journal Article (Journal Article)

We study enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the flow amplitude is large enough. The necessary and suffcient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow. In particular, we find that weakly mixing flows always enhance dissipation in this sense. The proofs are based on a general criterion for the decay of the semigroup generated by an operator of the form Γ + iAL with a negative unbounded self-adjoint operator Γ, a self-adjoint operator L, and parameter A » 1. In particular, they employ the RAGE theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian (related to a classical theorem of Wiener on Fourier transforms of measures). Applications to quenching in reaction-diffusion equations are also considered.

Full Text

Duke Authors

Cited Authors

  • Constantin, P; Kiselev, A; Ryzhik, L; ZlatoŇ°, A

Published Date

  • January 1, 2008

Published In

Volume / Issue

  • 168 / 2

Start / End Page

  • 643 - 674

International Standard Serial Number (ISSN)

  • 0003-486X

Digital Object Identifier (DOI)

  • 10.4007/annals.2008.168.643

Citation Source

  • Scopus