Enhanced convergence rates and asymptotics for a dispersive Boussinesq-type system with large ill-prepared data

Abstract

We obtain, for a stratified, rotating, incompressible Navier–Stokes system, generalized asymptotics as the Rossby number $𝜀$ goes to zero (without assumptions on the diffusion coefficients). For ill-prepared, less regular initial data with large blowing-up norm in terms of $𝜀$, we show global well-posedness and improved convergence rates (as a power of $𝜀$) towards the solution of the limit system, called the 3-dimensional quasigeostrophic system. Aiming for significant improvements required us to avoid as much as possible resorting to classical energy estimates involving oscillations. Our approach relies on the use of structures and symmetries of the limit system, and of highly improved Strichartz-type estimates.