Diffusion and Mixing in Fluid Flow

TL;DR

This paper characterizes flows that significantly enhance diffusive mixing on manifolds, linking spectral properties of the flow to rapid dissipation, with applications to reaction-diffusion systems.

Contribution

It provides a sharp spectral criterion for flows that accelerate mixing arbitrarily fast, extending understanding of dissipation enhancement in fluid dynamics.

Findings

Weakly mixing flows always enhance dissipation.

Spectral properties determine the effectiveness of mixing.

Applications include improved quenching in reaction-diffusion equations.

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 sufficient 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 G+iAL with a negative unbounded self-adjoint operator G, 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.