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

TL;DR

This paper proves an improved lower bound on the lifespan of solutions to 2D ideal MHD equations with small initial magnetic fields, using Els"asser formulation and Besov spaces, revealing solutions can persist longer than standard theory suggests.

Contribution

It introduces a novel approach combining Els"asser formulation and endpoint Besov spaces to extend the lifespan analysis of ideal MHD solutions in two dimensions.

Findings

Lifespan of solutions tends to infinity as initial magnetic field size approaches zero.

Equivalence between original and Els"asser formulations is established for a broad class of weak solutions.

Counterexamples demonstrate the sharpness of the assumptions made.

Abstract

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.