Regularization by noise for the inviscid primitive equations

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.