理想MHD的Elsässer公式和两个空间维度的改进寿命
Elsässer formulation of the ideal MHD and improved lifespan in two space dimensions
摘要：
本文给出了空间维$d=2$情形下理想MHD方程解的寿命的一个改进的下界。特别地，对于大小（比如）为$\varepsilon>0$的小初始磁场$b_0$，相应解的寿命$T_\varepsilon>0$在极限$\varepsilon\rightarrow0^+$中变为$+\infty$。这样的结果并不遵循标准的准线性双曲理论。为了证明这一点，有三个关键要素：首先，在端点Besov空间$B^s_{\infty,r}$中，在$s>1$和$r\in[1,+\infty]$或$s=r=1$的条件下工作;此外，使用理想MHD的Elsässer公式，在其涡量公式中重新转换;最后，利用非线性项的特殊结构。我们还严格地建立了一个大类的弱解的理想MHD和Elsässer制定的原始配方之间的等价性。明确的反例的建设表明我们的假设的尖锐性。相关的非唯一性问题也进行了讨论。
英文摘要：
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 Elsässer 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 Elsässer 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.
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