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

Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics