论文标题
在两个空间维度中的理想MHD和改善寿命的Elsässer公式
Elsässer formulation of the ideal MHD and improved lifespan in two space dimensions
论文作者
论文摘要
在本文中，在空间尺寸$ d = 2 $的情况下，我们显示了理想MHD方程解决方案的寿命的改进的下限。特别是，对于小初始磁场，$ b_0 $的大小（例如）$ \ varepsilon> 0 $，lifespan $ t_ \ varepsilon> 0 $的相应解决方案的$ to $+\ infty $ in Limit of限制$ \ VAREPSILON \ VAREPSILON \ rightarrow0^+$。 这样的结果并非源于标准的准线性双曲理论。为了证明这一点，三种是至关重要的成分：首先，在条件$ s> 1 $和$ r \ in [1，+\ iffty] $或$ s = r = r = 1 $的情况下，在端点上工作$ b^s _ {\ infty，r} $；此外，使用理想MHD的Elsässer配方，以其涡度配方重新铸造；最后，利用非线性术语的特殊结构。 我们还严格地建立了理想MHD的原始配方与其Elsässer公式之间的等效性，用于大量的弱解决方案。明确的反例的构建显示了我们假设的清晰度。还讨论了相关的非唯一性问题。
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.
