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

TL;DR

This paper proves that primitive equations with certain rough transport noise are globally well-posed and exhibit instant regularization, using novel anisotropic Besov spaces and $L^q$-techniques, extending understanding of turbulence modeling.

Contribution

It introduces new anisotropic Besov spaces tailored to primitive equations and demonstrates global well-posedness and regularization under rough noise conditions, including Kraichnan's type.

Findings

Global well-posedness for primitive equations with rough transport noise.

Identification of critical anisotropic Besov spaces for these equations.

Instantaneous regularization results for $H^1$-data with $ extgamma > 1$.

Abstract

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 $.