Stable Singularity Formation for the Inviscid Primitive Equations

TL;DR

This paper characterizes two distinct mechanisms leading to finite-time singularities in a reduced model of the inviscid primitive equations, enhancing understanding of blowup phenomena in geophysical fluid dynamics.

Contribution

It provides a comprehensive description of two stable blowup mechanisms for a reduced PDE related to the inviscid primitive equations, including detailed profiles and stability analysis.

Findings

First mechanism involves boundary-localized shock formation with a logarithmic blow-up law.

Second mechanism features a non-smooth profile with pressure effects negligible.

Both mechanisms are stable under specific smoothness conditions.

Abstract

The primitive equations (PEs) model large scale dynamics of the oceans and the atmosphere. While it is by now well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blowup mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic in time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameters family of non-smooth profiles, and is stable only by smoother perturbations.