Sharp decay estimates for global solutions to the incompressible rotating Navier--Stokes equations

Abstract

In this paper, we consider the three-dimensional incompressible rotating Navier--Stokes equations and establish the sharp $L^p$ decay estimates of global solutions. We reveal that the optimal $L^p$ decay rates for $2<p<\infty$ are strictly faster than those obtained in existing results by interpolation between the $L^2$ unitary identity and $L^\infty$ dispersive estimates, although the endpoint cases were known to be sharp. Moreover, the optimality of decay rates is also proved by the lower bound estimate for a specific initial datum. The underlying mechanism lies in the anisotropic degeneracy of the oscillatory integrals arising from the Coriolis force.