Analyticity in space and time for global solutions to the anisotropic Navier--Stokes equations in the critical $L^p(\mathbb{R}^3)$ framework

TL;DR

This work proves space-time analyticity for global solutions to the 3D incompressible anisotropic Navier–Stokes equations with horizontal viscosity in the critical $L^p$ Besov framework. It combines an anisotropic Littlewood–Paley setup with time–space analytic weights and a carefully designed auxiliary function $f_M(t)$ to manage vertical derivative loss and propagate analyticity. The authors establish linear and nonlinear analytic a priori estimates in scaling-critical anisotropic Besov spaces, handle the pressure term via a decomposition, and construct global solutions through a frequency-truncated approximation and compactness, with uniqueness in the regime $1\le p\le 2$. The results extend the $L^p$-based well-posedness theory for anisotropic Navier–Stokes by achieving real analyticity in time and all spatial variables without assuming vertical analyticity of $u_{0,3}$, thereby advancing the understanding of geophysical-fluid-like models with dominant horizontal dissipation.

Abstract

In the present paper, we consider the real analyticity of the global solutions to the $3$D incompressible anisotropic Navier--Stokes equations. We show that if only the horizontal component of initial velocity is small and analytic in $x_3$, then there exists a unique global solution which is analytic in $t>0$ and $x\in \mathbb{R}^3$. Our functional framework lies in some anisotropic Besov spaces based on $L^p(\mathbb{R}^3)$. To our best knowledge, this paper is the first contribution to the well-posedness of the anisotropic Navier--Stokes equations in function spaces of the Besov type based on the full $L^p(\mathbb{R}^3)$ setting.

Theorems & Definitions (25)

• Theorem 1.1
• Remark 1.2
• Definition 2.1
• Lemma 2.2
• Remark 2.3
• proof : Proof of Lemma \ref{['lemm:prod']}
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7