Anomalous Regularization in Kraichnan's Passive Scalar Model

Abstract

We consider the advection of a passive scalar by a divergence free random Gaussian field, white in time and Hölder 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.

25 pages. Probab. Theory Relat. Fields (2026)

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