Global uniform regularity and vanishing vertical viscosity limit for the compressible Navier--Stokes equations in the half-space

Abstract

In geophysical flows such as large-scale ocean dynamics, the vertical viscosity is often much smaller than the horizontal viscosity. This anisotropy makes it natural to ask whether solutions of the full anisotropic compressible Navier--Stokes equations converge, as the vertical viscosity coefficient $\varepsilon \to 0$, to solutions of a horizontally dissipative limit system, and whether this limit can be justified globally in time. Prior work has answered this question locally in time or in the incompressible setting. We resolve this problem for the three-dimensional compressible Navier--Stokes equations in the upper half-space with the Navier slip boundary condition. This paper establishes two main results for small perturbations of a constant equilibrium state. First, we prove the existence of a unique global-in-time solution whose conormal Sobolev norm remains uniformly bounded for all $t \ge 0$ and all $\varepsilon \in (0,1)$. Second, we justify the global vanishing vertical viscosity limit. More precisely, we show that the solutions converge to a global solution of the horizontally dissipative compressible Navier--Stokes system. This provides the first rigorous justification of the anisotropic viscosity limit for compressible flows on an infinite time interval.