Els\"asser formulation of the ideal MHD and improved lifespan in two space dimensions

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.