Large Time Behavior for Solutions to the Anisotropic Navier–Stokes Equations in a 3D Half-space.

This article investigates the large-time behavior of solutions to the anisotropic incompressible Navier–Stokes equations in a three-dimensional half-space, focusing on the case where vertical viscosity is negligible compared to horizontal viscosity—a model relevant to geophysical fluid dynamics. The main results establish the global existence and uniqueness of small solutions in suitable anisotropic Besov-type Chemin–Lerner spaces and provide optimal decay estimates in anisotropic Lebesgue norms. Notably, the study verifies an enhanced dissipation mechanism for the vertical component of the velocity field, which decays faster (like a 3D heat kernel) than the horizontal components (which decay like a 2D heat kernel). The analysis overcomes challenges posed by boundary conditions and nonlocal operators by employing Littlewood–Paley theory and introducing appropriate function spaces, and it further proves uniform boundedness of solutions in the \(L^1\)-norm under stronger initial data assumptions.