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

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.