Global Small Analytic Solution of 3-D Anisotropic Navier-Stokes System

In this paper, we prove the global existence of analytic solution for 3D anisotropic Navier-Stokes system with initial data which is small and analytic in the vertical variable. We shall also prove that this solution will be analytic in the horizontal variables soon after \(t>0.\) $t>0.$ Furthermore, we show that the ratio between the analytic radius, \(R_\textrm{h}(t),\) $R_{\text{h}}(t),$ of the solution in the horizontal variables and \( \sqrt{t}\) $\sqrt{t}$ satisfies \(\lim _{t\rightarrow 0_+}\frac{R_\textrm{h}(t)}{\sqrt{t}}=\infty .\) ${lim}_{t\to 0_{+}}\frac{R_{\text{h}}(t)}{\sqrt{t}}=\infty .$