  In the present paper, we show an improved lower bound for the lifespan of the
solutions to the ideal MHD equations in the case of space dimension $d=2$. In
particular, for small initial magnetic fields $b_0$ of size (say)
$\varepsilon>0$, the lifespan $T_\varepsilon>0$ of the corresponding solution
goes to $+\infty$ in the limit $\varepsilon\rightarrow0^+$.
  Such a result does not follow from standard quasi-linear hyperbolic theory.
For proving it, three are the crucial ingredients: first of all, to work in
endpoint Besov spaces $B^s_{\infty,r}$, under the condition $s>1$ and
$r\in[1,+\infty]$ or $s=r=1$; moreover, to use the Els\"asser formulation of
the ideal MHD, recasted in its vorticity formulation; finally, to take
advantage of the special structure of the non-linear terms.
  We also rigorously establish the equivalence between the original formulation
of the ideal MHD and its Els\"asser formulation for a large class of weak
solutions. The construction of explicit counterexamples shows the sharpness of
our assumptions. Related non-uniqueness issues are discussed as well.
