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Anomalous diffusion by fractal homogenization

Scott Armstrong, Vlad Vicol

TL;DR

The authors construct an explicit deterministic, time-space periodic, incompressible vector field with fractal (multi-scale) structure built from alternating shear flows. Using a fractal (renormalization) homogenization framework, they recursively define scale-dependent correctors and renormalized diffusivities, proving that the effective diffusivity becomes order one on the macroscopic scale. This leads to anomalous dissipation of scalar variance: as κ → 0, the dissipation κ∥∇θ^κ∥^2 remains bounded away from zero for mean-zero initial data. The approach provides a rigorous, scale-by-scale mechanism for anomalous diffusion in a deterministic setting and offers a quantitative homogenization methodology applicable to turbulent-like transport problems.

Abstract

For every $α< \frac13$, we construct an explicit divergence-free vector field $\mathbf{b}(t,x)$ which is periodic in space and time and belongs to $C^0_t C^α_x \cap C^α_t C^0_x$ such that the corresponding scalar advection-diffusion equation $$\partial_t θ^κ+ \mathbf{b} \cdot \nabla θ^κ- κΔθ^κ= 0$$ exhibits anomalous dissipation of scalar variance for arbitrary $H^1$ initial data: $$\limsup_{κ\to 0} \int_0^{1} \int_{\mathbb{T}^d} κ\| \nabla θ^κ(t,x) \|^2 \,dx\,dt >0.$$ The vector field is deterministic and has a fractal structure, with periodic shear flows alternating in time between different directions serving as the base fractal. These shear flows are repeatedly inserted at infinitely many scales in suitable Lagrangian coordinates. Using an argument based on ideas from quantitative homogenization, the corresponding advection-diffusion equation with small $κ$ is progressively renormalized, one scale at a time, starting from the (very small) length scale determined by the molecular diffusivity up to the macroscopic (unit) scale. At each renormalization step, the effective diffusivity is enhanced by the influence of advection on that scale. By iterating this procedure across many scales, the effective diffusivity on the macroscopic scale is shown to be of order one.

Anomalous diffusion by fractal homogenization

TL;DR

The authors construct an explicit deterministic, time-space periodic, incompressible vector field with fractal (multi-scale) structure built from alternating shear flows. Using a fractal (renormalization) homogenization framework, they recursively define scale-dependent correctors and renormalized diffusivities, proving that the effective diffusivity becomes order one on the macroscopic scale. This leads to anomalous dissipation of scalar variance: as κ → 0, the dissipation κ∥∇θ^κ∥^2 remains bounded away from zero for mean-zero initial data. The approach provides a rigorous, scale-by-scale mechanism for anomalous diffusion in a deterministic setting and offers a quantitative homogenization methodology applicable to turbulent-like transport problems.

Abstract

For every , we construct an explicit divergence-free vector field which is periodic in space and time and belongs to such that the corresponding scalar advection-diffusion equation exhibits anomalous dissipation of scalar variance for arbitrary initial data: The vector field is deterministic and has a fractal structure, with periodic shear flows alternating in time between different directions serving as the base fractal. These shear flows are repeatedly inserted at infinitely many scales in suitable Lagrangian coordinates. Using an argument based on ideas from quantitative homogenization, the corresponding advection-diffusion equation with small is progressively renormalized, one scale at a time, starting from the (very small) length scale determined by the molecular diffusivity up to the macroscopic (unit) scale. At each renormalization step, the effective diffusivity is enhanced by the influence of advection on that scale. By iterating this procedure across many scales, the effective diffusivity on the macroscopic scale is shown to be of order one.
Paper Structure (38 sections, 36 theorems, 639 equations)

This paper contains 38 sections, 36 theorems, 639 equations.

Table of Contents

  1. Introduction
  2. Motivation and prior results on anomalous dissipation of scalar variance
  3. An outline of the proof: fractal homogenization
  4. Uniform estimates for the scalar
  5. Notation
  6. Outline of the paper
  7. The fractal vector field: construction and regularity
  8. A list of the ingredients used in the construction
  9. Construction of the vector field
  10. Regularity of the stream functions and associated flows
  11. Material derivative estimates
  12. Correctors and renormalized diffusivities
  13. The correctors: definitions and estimates
  14. Correctors and effective diffusivity for smoothly alternating shear flows
  15. The renormalized diffusivities: recurrence, averaging and estimates
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $d\geq 2$ and $\alpha \in (0,1/3)$. There exists a vector field which satisfies e.divfree such that, for every mean-zero initial datum $\theta_0 \in H^1(\mathbb{T}^d)$, the family of unique solutions $\{ \theta^\kappa\}_{\kappa>0} \in C([0,1];L^2(\mathbb{T}^d))$ of the advection-diffusion equation e.passive.scalar satisfy for some constant $\varrho=\varrho(d,\theta_0)\in(0,1]$ which depends

Theorems & Definitions (76)

  • Theorem 1.1: Anomalous dissipation of scalar variance
  • Remark 2.1: Convention for the constants
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['p.SAMS.regularity']}
  • Corollary 2.3
  • proof : Proof of Corollary \ref{['c.phim']}
  • Corollary 2.4
  • proof : Proof of Corollary \ref{['c.flowreg']}
  • Proposition 2.5
  • proof : Proof of Proposition \ref{['p.material.goal']}
  • ...and 66 more