Global Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation

Although the classical Yudovich theory guarantees global existence and uniqueness of two-dimensional Euler solutions with bounded vorticity, their higher-order norms may still grow in time. This raises a fundamental question: how much dissipation we need to ensure global stability and decay? While the full Laplacian dissipation is sufficient, the stabilizing effect of weaker, anisotropic dissipation is considerably more subtle. In this paper, we investigate the global stability and long-time decay of the two-dimensional incompressible Navier-Stokes equations with fractional horizontal dissipation $Λ_1^{2s}u$, where $s<1$. A central challenge is to recover sufficient decay of the second velocity component, whose weak dissipation obstruct the closure of conventional anisotropic energy estimates. Previous work by Wang-Wu-Zhu [arXiv:2601.16110] resolved this difficulty whereas the spatial weighted energy functionals must be used when $s$ approaches $1$. The main purpose of our manuscript is to remove this restriction. We introduce a hierarchy of carefully designed time-weighted anisotropic energy functionals which captures the interplay between horizontal fractional dissipation and the decay of velocity components. Under such initial assumptions, the characteristics of anisotropic dissipation is presented more straightforwardly, which also reveals the more essential intrinsic structure of fluid stability.