Anomalous regularization for the non-smooth Kraichnan and Kazantsev-Kraichnan models on the torus

We examine Kraichnan’s passive scalar model on the $d$-dimensional torus. On $\mathbb{R}^{d}$, this model exhibits anomalous dissipation and anomalous regularization. We establish that the model on the torus enjoys a similar anomalous regularization property. The added difficulty in this setting is the absence of translation invariance, and to get around this, we use a scheme relying on an expansion of the correlation function of the driving noise. We establish a similar result for the Kazantsev-Kraichnan model on the torus. Based on joint work with Lucio Galeati and Mario Maurelli.