  We give lower bounds for the lifespan of a solution to the inviscid
Boussinesq system. In dimension two, we point out that it tends to infinity
when the initial (relative) temperature tends to zero. This is, to the best of
our knowledge, the first result of this kind for the inviscid Boussinesq
system. In passing, we provide continuation criteria (of independent interest)
in the $N$-dimensional case. In the second part of the paper, our method is
adapted to handle the axisymmetric incompressible Euler equations with swirl.
