Long-time solvability for the 2D inviscid Boussinesq equations with borderline regularity and dispersive effects

We are concerned with the long-time solvability for 2D inviscid Boussinesq equations for a larger class of initial data which covers the case of borderline regularity. First we show the local solvability in Besov spaces uniformly with respect to a parameter $\kappa$ associated with the stratification of the fluid. Afterwards, employing a blow-up criterion and Strichartz-type estimates, the long-time solvability is obtained for large $\kappa$ regardless of the size of initial data.