Anomalous Regularization in Kazantsev-Kraichnan Model

TL;DR

This paper studies a passive vector field in a turbulent flow model, showing that regularization effects persist in certain regimes, with implications for fluid dynamics and magnetohydrodynamics.

Contribution

It demonstrates that regularization properties of scalar advection extend to vector fields in the Kazantsev-Kraichnan model under specific conditions.

Findings

Regularization holds in dimensions d≥3.

Implications for 3D Euler equations with stochastic turbulence.

Relevance to magnetohydrodynamic turbulence.

Abstract

This work investigates a passive vector field which is transported and stretched by a divergence-free Gaussian velocity field, delta-correlated in time and poorly correlated in space (spatially nonsmooth). Although the advection of a scalar field (Kraichnan's passive scalar model) is known to enjoy regularizing properties, the potentially competing stretching term in vector advection may induce singularity formation. We establish that the regularization effect is actually retained in certain regimes. While this is true in any dimension $d\ge 3$, it notably implies a regularization result for linearized 3D Euler equations with stochastic modeling of turbulent velocities, and for the induction equation in magnetohydrodynamic turbulence.