Diusion and mixing in fluid flow

Peter Constantin, A. Kiselev, L. Ryzhik, A. Zlatoš

Research output: Contribution to journalArticle

81 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)643-674
Number of pages32
JournalAnnals of Mathematics
Volume168
Issue number2
DOIs
StatePublished - 2008

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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