Regularization by noise for the inviscid primitive equations

TL;DR

This paper shows that adding appropriate random noise to the inviscid primitive equations can restore well-posedness and prevent finite-time blowups, which are issues in the deterministic case.

Contribution

It introduces stochastic regularization techniques that ensure local and global well-posedness of the primitive equations under random perturbations.

Findings

Random noise restores local well-posedness in Gevrey classes.

Random damping prevents finite-time singularities.

Solutions exist globally with high probability under stochastic effects.

Abstract

The deterministic inviscid primitive equations (also called the hydrostatic Euler equations) are known to be ill-posed in Sobolev spaces and in Gevrey classes of order strictly greater than 1, and some of their analytic solutions exist only locally in time and exhibit finite-time blowup. This work demonstrates that introducing suitable random noise can restore the local well-posedness and prevent finite-time blowups. Specifically, random diffusion addresses the ill-posedness in certain Gevrey classes, allowing us to establish the local well-posedness almost surely and the global existence of solutions with high probability. In the case of random damping (linear multiplicative noise), the noise prevents analytic solutions from forming singularities in finite time, resulting in globally existing solutions with high probability.