Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
14
The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga