Analyticity and optimal decay estimate for the equatorial ocean flow model

In this paper, we study a nonlocal evolution equation modeling oceanic flows in equatorial regions. The equation involves quadratic nonlinearities with the Hilbert transform, reflecting intrinsic nonlocal wave-current interactions in equatorial dynamics. We establish global well-posedness for small initial data in homogeneous Sobolev spaces. Furthermore, solutions are shown to be spatially analytic for all positive times, and optimal time-decay estimates are obtained. These results demonstrate that, despite the presence of genuinely nonlocal Hilbert-type nonlinearities, the equation enjoys strong regularizing and dissipative properties.