Anomalous properties of 2D active scalars perturbed by rough transport noise
Abstract
We study SPDEs associated with $2$D active scalars driven by incompressible transport noise of Kraichnan type, with regularity exponent α∈ (0,1); our examples include the Euler, SQG and IPM systems. We investigate whether a number of ``turbulent'' phenomenologies, which are well understood in the linear Kraichnan model, persist in this nonlinear setting, uniformly in vanishing viscosity approximations. First, for suitable values of α and initial data in Lpx, we establish anomalous regularization estimates, measured in appropriate endpoint Besov-type spaces of regularity β=β(α,p)>0. Remarkably, these results allow for some scaling supercritical regimes of the parameters α,p; on the other hand, for (sub)critical parameters, we recover the same regularity exponent β=1−α as in the linear case. Second, in the (sub)critical case, we further prove anomalous integrability, namely solutions becoming instantaneously L∞x-valued at positive times, uniformly in the viscosity; moreover, in this case we establish strong existence and pathwise uniqueness of solutions to the inviscid SPDE, which are recovered as the unique vanishing viscosity limit. Finally, in the $2$D Euler case, for α∈ (0,1/2), we establish anomalous dissipation of enstrophy and sharpness of anomalous regularization.
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BibTeX
@misc{galeati2026anomalous,
title = {{Anomalous properties of 2D active scalars perturbed by rough transport noise}},
author = {Lucio Galeati and Eliseo Luongo and Umberto Pappalettera},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25897},
eprint = {2609.25897},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25897},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Anomalous properties of 2D active scalars perturbed by rough transport noise
AU - Lucio Galeati
AU - Eliseo Luongo
AU - Umberto Pappalettera
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25897
UR - https://arxiv.org/abs/2609.25897
AB - We study SPDEs associated with $2$D active scalars driven by incompressible transport noise of Kraichnan type, with regularity exponent α∈ (0,1); our examples include the Euler, SQG and IPM systems. We investigate whether a number of ``turbulent'' phenomenologies, which are well understood in the linear Kraichnan model, persist in this nonlinear setting, uniformly in vanishing viscosity approximations. First, for suitable values of α and initial data in L^p_x, we establish anomalous regularization estimates, measured in appropriate endpoint Besov-type spaces of regularity β=β(α,p)>0. Remarkably, these results allow for some scaling supercritical regimes of the parameters α,p; on the other hand, for (sub)critical parameters, we recover the same regularity exponent β=1−α as in the linear case. Second, in the (sub)critical case, we further prove anomalous integrability, namely solutions becoming instantaneously L^∞_x-valued at positive times, uniformly in the viscosity; moreover, in this case we establish strong existence and pathwise uniqueness of solutions to the inviscid SPDE, which are recovered as the unique vanishing viscosity limit. Finally, in the $2$D Euler case, for α∈ (0,1/2), we establish anomalous dissipation of enstrophy and sharpness of anomalous regularization.
ER -