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This work is the continuation of the recent paper \cite{D2} devoted to thedensity-dependent incompressible Euler equations. Here we concentrate on thewell-posedness issue in Besov spaces of type $B^s_{\infty,r}$ embedded in theset of Lipschitz continuous functions, a functional framework which containsthe particular case of H\"older spaces and of the endpoint Besov space$B^1_{\infty,1}.$ For such data and under the nonvacuum assumption, weestablish the local well-posedness and a continuation criterion in the spiritof that of Beale, Kato and Majda in \cite{BKM}.  In the last part of the paper, we give lower bounds for the lifespan of asolution. In dimension two, we point out that the lifespan tends to infinitywhen the initial density tends to be a constant. This is, to our knowledge, thefirst result of this kind for the density-dependent incompressible Eulerequations.

The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces