Well-posedness and ill-posedness of the primitive equations with fractional horizontal dissipation

TL;DR

This paper investigates the well-posedness of the two-dimensional primitive equations with fractional horizontal dissipation, identifying a critical dissipation exponent that determines stability and establishing conditions for global well-posedness.

Contribution

It precisely characterizes the transition between ill-posedness and well-posedness based on the fractional dissipation exponent and initial data, advancing understanding of geophysical fluid models.

Findings

Sharp transition at dissipation exponent α=1 between well-posedness and ill-posedness.

Global well-posedness established for dissipation α≥6/5.

The transition depends on initial data size and viscosity coefficient.

Abstract

The primitive equations (PE) are a fundamental model in geophysical fluid dynamics. While the viscous PE are globally well-posed, their inviscid counterparts are known to be ill-posed. In this paper, we study the two-dimensional incompressible PE with fractional horizontal dissipation. We identify a sharp transition between local well-posedness and ill-posedness at the critical dissipation exponent $\alpha = 1$. In the critical regime, this dichotomy exhibits a new phenomenon: the transition depends delicately on the balance between the size of the initial data and the viscosity coefficient. Our results precisely quantify the horizontal dissipation required to transition from inviscid instability to viscous regularity. We also establish a global well-posedness theory to the fractional PE, with sufficient dissipation $\alpha\geq\frac65$.