The primitive equations with rough transport noise: Global well-posedness and regularity

In this paper we establish global well-posedness and instantaneous regularization results for the primitive equations with transport noise of H\"{o}lder regularity $ \gamma>\frac{1}{2}$. It is known that if $\gamma<1$, then the noise is too rough for a strong formulation of primitive equations in an $L^2$-based setting. To handle rough noise, we crucially use $L^q$-techniques with $q> 2$. Interestingly, we identify a family of critical anisotropic Besov spaces for primitive equations, which is new even in the deterministic case. The behavior of these spaces reflects the intrinsic anisotropy of the primitive equations and plays an essential role in establishing global well-posedness and regularization. Our results cover Kraichnan's type noise with correlation greater than one, and as a by-product, a 2D noise reproducing the Kolmogorov spectrum of turbulence. Moreover, the instantaneous regularization is new also in the widely studied case of $H^1$-data and $\gamma>1 $.