We consider the generalised Surface Quasi-Geostrophic (gSQG) equations in \(\mathbb {R}^2\) with parameter \(\beta \in (0,1)\) , an active scalar model interpolating between SQG ( \(\beta =1\) ) and the 2D Euler equations ( \(\beta =0\) ) in vorticity form. Existence of weak \((L^1\cap L^p)\) -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 \(\theta _0\in L^1\cap L^p\) , for suitable values \(p\in [2,\infty ]\) related to the regularity degree \(\alpha \) of the noise and the singularity degree \(\beta \) of the velocity field; in particular, we can cover any \(\beta \in (0,1)\) for suitable \(\alpha \) and p and we can reach a suitable (“critical”) threshold. The result also holds in the presence of external forcing \(f\in L^1_t (L^1\cap L^p)\) 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 well-posedness for two-dimensional linear transport equation with random drift, with the same noise.
