论文标题
准同质理想MHD系统的严格衍生和良好的定位
Rigorous derivation and well-posedness of a quasi-homogeneous ideal MHD system
论文作者
论文摘要
本文的目标是双重的。一方面，我们介绍了经典理想MHD系统的准同质版本，并研究了其在关键的Besov空间中的功能良好$ r \ in [1，+\ infty] $，或$ s = 1+d/p $和$ r = 1 $。一个关键要素是系统\ textsl {via}的重新制定，所谓的elsässer变量。另一方面，在理想和耗散情况下，我们给出了准同质MHD模型的严格理由：当$ d = 2 $时，我们将从coriolis force的非均匀不可压缩的MHD系统中得出它们，在低rossby数字的状态下，对于一个恒定状态的小密度变化，我们将从coriolis force中。我们的证明方法依赖于原始系统的相对熵不等式，并根据初始数据的大小，Rossby数字的顺序以及粘度和电阻率系数的规律性，从而产生了精确的收敛速率。
The goal of this paper is twofold. On the one hand, we introduce a quasi-homogeneous version of the classical ideal MHD system and study its well-posedness in critical Besov spaces $B^s_{p,r}(\mathbb{R}^d)$, $d\geq2$, with $1<p<+\infty$ and under the Lipschitz condition $s>1+d/p$ and $r\in[1,+\infty]$, or $s=1+d/p$ and $r=1$. A key ingredient is the reformulation of the system \textsl{via} the so-called Elsässer variables. On the other hand, we give a rigorous justification of quasi-homogeneous MHD models, both in the ideal and in the dissipative cases: when $d=2$, we will derive them from a non-homogeneous incompressible MHD system with Coriolis force, in the regime of low Rossby number and for small density variations around a constant state. Our method of proof relies on a relative entropy inequality for the primitive system, and yields precise rates of convergence, depending on the size of the initial data, on the order of the Rossby number and on the regularity of the viscosity and resistivity coefficients.
