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∈ L¹ₜ (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.