Global weak solutions to a two-dimensional compressible MHD equations of viscous non-resistive fluids
Global weak solutions to a two-dimensional compressible MHD equations of viscous non-resistive fluids
复制标题
粘性无阻流体二维可压缩MHD方程的全局弱解
DOI：
10.1016/j.jde.2019.04.024
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发表时间：
2019
影响因子：
2.4
通讯作者：
Sun Yongzhong
中科院分区：
文献类型：
--
作者：
Li Yang;Sun Yongzhong
我们考虑了一个描述粘性、可压缩和导电流体在无电阻率垂直磁场作用下的演化的二维MHD模型。对任意绝热指数γ≥1建立了整体弱解的存在性。受文[15]中提出的近似格式的启发，我们考虑了一个具有人工扩散和压力项的两能级近似系统。在第一个水平，我们证明了正则化系统的全局适定性，并建立了正则解的Uniform-in-ε估计。在第二个水平，我们通过将ε设为0并推导出Uniform-in-δ估计，证明了具有人工压力的系统整体弱解的存在性。然后通过消除δ来构造原系统的整体弱解。极限通过中的关键问题是密度和磁场的近似序列的强收敛。这是通过遵循[15]、[26]中开发的技术，并使用Vasseur等人[33]在可压缩双流体模型的背景下开发的变量约简的新技术来实现的，以便处理交叉项。
We consider a two-dimensional MHD model describing the evolution of viscous, compressible and electrically conducting fluids under the action of vertical magnetic field without resistivity. Existence of global weak solutions is established for any adiabatic exponent γ≥ 1. Inspired by the approximate scheme proposed in [15], we consider a two-level approximate system with artificial diffusion and pressure term. At the first level, we prove global well-posedness of the regularized system and establish uniform-in-ε estimates to the regular solutions. At the second level, we show global existence of weak solutions to the system with artificial pressure by sending ε to 0 and deriving uniform-in-δ estimates. Then global weak solution to the original system is constructed by vanishing δ. The key issue in the limit passage is the strong convergence of approximate sequence of the density and magnetic field. This is accomplished by following the technique developed in [15],[26] and using the new technique of variable reduction developed by Vasseur et al.[33] in the context of compressible two-fluid model so as to handle the cross terms.