Regularization by rough Kraichnan noise for the generalised SQG equations

We consider the generalised Surface Quasi-Geostrophic (gSQG) equations in R2 with parameter β ∈ (0, 1), an active scalar model interpolating between SQG (β = 1) and the 2D Euler equations (β = 0) in vorticity form. Existence of weak (L1 ∩ Lp)-valued solutions in the deterministic setting is known, but their uniqueness is open. We show that the addition of a rough Stratonovich transport noise of Kraichnan type regularizes the PDE, providing strong existence and pathwise uniqueness of solutions for initial data θ0 ∈ L1 ∩ Lp, for suitable values p ∈ [2,∞] related to the regularity degree α of the noise and the singularity degree β of the velocity field; in particular, we can cover any β ∈ (0, 1) for suitable α and p and we can reach a suitable (“critical”) threshold. The result also holds in the presence of external forcing f ∈ L1 t (L1 ∩ Lp) and solutions are shown to depend continuously on the data of the problem; furthermore, they are well approximated by vanishing viscosity and regular approximations. With similar techniques, we also show wellposedness for two-dimensional linear transport equation with random drift, with the same noise.