Local and Global Solutions of the 3D-NSE in Homogeneous Lei–Lin–Gevrey Spaces

This paper investigates the existence and uniqueness of local and global solutions to the incompressible three-dimensional Navier–Stokes equations within the framework of homogeneous Lei–Lin–Gevrey spaces Xa,γρ(R3), where ρ∈[−1,0),a>0, and γ∈(0,1). These function spaces combine the critical scaling structure of the Lei–Lin spaces with the exponential regularity of Gevrey classes, thereby enabling a refined treatment of analytic regularity and frequency localization. The main results are obtained under the assumption of small initial data in the critical Lei–Lin space Xρ(R3), extending previous works and improving regularity thresholds. In particular, we establish that for suitable initial data, the Navier–Stokes system admits unique solutions globally in time. The influence of the Gevrey parameter γ on the high-frequency behavior of solutions is also discussed. This work contributes to a deeper understanding of regularity and decay properties in critical and supercritical regimes.