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 paper proves that under certain initial conditions, solutions to the 3D anisotropic Navier--Stokes equations are globally analytic in space and time within an $L^p$-based anisotropic Besov space framework.

Contribution

It establishes the first well-posedness results for anisotropic Navier--Stokes equations in Besov spaces based on the full $L^p( eal^3)$ setting, considering anisotropic analyticity.

Findings

Existence of unique global solutions under small horizontal initial velocity.

Solutions are analytic in time and space for positive time.

First results of this kind in anisotropic Besov spaces based on $L^p( eal^3)$.

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.