Global well-posedness of the 2D primitive equations with fractional horizontal dissipation

TL;DR

This work analyzes the 2D primitive equations with fractional horizontal dissipation $\Lambda_h^\alpha$ and hydrostatic balance, establishing global well-posedness for strong solutions when $\alpha\ge\alpha_0\approx1.1108$ for large initial data, and for $\alpha\in[1,\alpha_0)$ with small hydrostatic vorticity $\omega_0$ under a finite initial-regularity condition. The authors develop a hierarchy of enhanced a priori energy bounds, leveraging an $L^{\infty}$ bound on $\omega$ and iterative regularity upgrades to propagate control from $u$ to $\omega$ and its derivatives. A Beale–Kato–Majda type criterion is verified to guarantee uniqueness, and a refined, data-dependent regularity threshold $\delta_m$ is identified for the initial velocity to ensure global well-posedness. The results extend prior global well-posedness thresholds from $\alpha\ge 6/5$ down to $\alpha_0$, clarifying the role of horizontal dissipation in the geophysical primitive equations and illustrating a sharp transition at $\alpha=1$ between ill-posedness and well-posedness in the critical regime. The analysis relies on fractional Leibniz rules, interpolation inequalities, anisotropic estimates, and an iterative energy framework that ties regularity improvements to bounds on hydrostatic vorticity.

Abstract

In this paper, we investigate the two-dimensional incompressible primitive equations with fractional horizontal dissipation. Specifically, we establish global well-posedness of strong solutions for arbitrarily large initial data when the dissipation exponent satisfies $α\geqα_{0}\approx1.1108$. In addition, we prove global well-posedness of strong solutions for small initial data when $α\in [1, α_0)$. Notably, the smallness assumption is imposed only on the $L^\infty$ norm of the initial vorticity.

Figures (2)

• Figure 1: An illustration of the iteration scheme. Left ($1<\alpha\leq\frac{4}{\sqrt{15}}$): the maximal enhancement on $\rho$ is $\rho_M$, which is smaller than $\rho^*$. Right ($\frac{4}{\sqrt{15}}<\alpha<\frac{6}{5}$): there is a path such that $\rho$ reaches $\rho^*$, and the minimal requirement on $\delta$ is $\delta_{**}$.
• Figure 2: An illustration on the admissible region, and the optimal choice of $(\rho,\delta)=(\rho^*, \delta_{**})$ .

Theorems & Definitions (17)

• Theorem 1.1
• Remark 1
• Theorem 1.2
• Remark 2
• Lemma 2.1: Fractional Leibniz rule
• Lemma 2.2: Interpolations
• Lemma 3.1
• Remark 3
• proof : Proof of Lemma \ref{['lem:uimprove']}
• Lemma 3.2