Anomalous Regularization in Kraichnan's Passive Scalar Model

TL;DR

This paper demonstrates that in Kraichnan's passive scalar model, the passive scalar field experiences an unexpected regularization effect, transforming initial distributions into smoother functions almost instantly due to the model's dynamics.

Contribution

It reveals an anomalous regularization phenomenon in the Kraichnan model, showing immediate smoothing of initial conditions through Sobolev norm analysis.

Findings

Negative Sobolev norms decrease rapidly

Initial distributions become regularized instantly

Anomalous smoothing effect observed in the model

Abstract

We consider the advection of a passive scalar by a divergence free random Gaussian field, white in time and H\"older regular in space (rough Kraichnan's model), a well established synthetic model of passive scalar turbulence. By studying the evolution of negative Sobolev norms, we show an anomalous regularization effect induced by the dynamics: distributional initial conditions immediately become functions of positive Sobolev regularity.