Anomalous properties of 2D active scalars perturbed by rough transport noise

Source abstract

We study SPDEs associated with $2$D active scalars driven by incompressible transport noise of Kraichnan type, with regularity exponent $α\in (0,1)$; 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 $L^\infty_x$-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 $α\in (0,1/2)$, we establish anomalous dissipation of enstrophy and sharpness of anomalous regularization.