Annals of Mathematical Sciences and Applications

Volume 2 (2017)

Number 1

Spectral analysis and computation of effective diffusivities in space-time periodic incompressible flows

Pages: 3 – 66



N. Benjamin Murphy (Department of Mathematics, University of California at Irvine)

Elena Cherkaev (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

Jack Xin (Department of Mathematics, University of California at Irvine)

Jingyi Zhu (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

Kenneth M. Golden (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)


The diffusive transport of passive tracers or particles can be enhanced by incompressible, turbulent flow fields. Analyzing the effective behavior is a challenging problem with theoretical and practical importance in many areas of science and engineering, ranging from the transport of mass, heat, and pollutants in geophysical flows to sea ice dynamics and turbulent combustion. The long time, large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity tensor $\mathsf{D}^{*}$. Two different formulations of integral representations for $\mathsf{D}^{*}$ were developed for the case of time-independent fluid velocity fields, involving spectral measures of bounded self-adjoint operators acting on vector fields and scalar fields, respectively. Here, we extend both of these approaches to the case of space-time periodic velocity fields, allowing for chaotic dynamics, providing rigorous integral representations for $\mathsf{D}^{*}$ involving spectral measures of unbounded self-adjoint operators.We prove the different formulations are equivalent. Their correspondence follows from a one-to-one isometry between the underlying Hilbert spaces. We also develop a Fourier method for computing $\mathsf{D}^{*}$, which captures the phenomenon of residual diffusion related to Lagrangian chaos of a model flow. This is reflected in the spectral measure by a concentration of mass near the spectral origin.


advective diffusion, homogenization, effective diffusivity, spectral measure, integral formula, Fourier method, generalized eigenvalue computation, residual diffusion

2010 Mathematics Subject Classification

Primary 47B15. Secondary 35C15, 65C60, 76B99, 76M22, 76M50, 76R99.

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Published 13 January 2017