SHARP DECAY ESTIMATES FOR AN ANISOTROPIC LINEAR SEMIGROUP AND APPLICATIONS TO THE SURFACE QUASI-GEOSTROPHIC AND INVISCID BOUSSINESQ SYSTEMS

At the core of this article is an improved, sharp dispersive estimate for the anisotropic linear semigroup e(R1t) arising in both the study of the dispersive surface quasi-geostrophic (SQG) equation and the inviscid Boussinesq system. We combine the decay estimate with a blow-up criterion to show how dispersion leads to long-time existence of solutions to the dispersive SQG equation, improving the results obtained using hyperbolic methods. In the setting of the inviscid Boussinesq system it turns out that linearization around a specific stationary solution leads to the same linear semigroup, so that we can make use of analogous techniques to obtain stability of the stationary solution for an increased time span.