We consider the Dirichlet problem for a compressible two-fluid model in multi-dimensions. It consists of the continuity equations for each fluid and the momentum equations for the mixture. This model can be derived from the compressible two-fluid model with equal velocities (Bresch et al., in Arch Rational Mech Anal 196:599–629, 2010) and from a scaling limit of the Vlasov-Fokker-Planck/compressible Navier-Stokes (Mellet and Vasseur, Commun Math Phys 281(3):573–596, 2008) (see also the compressible Oldroyd-B model with stress diffusion (Barrett et al., Commun Math Sci 15:1265–1323, 2017). Another interesting connection is that it is formally the equations of compressible magnetohydrodynamic (MHD) flows without resistivity in two dimensions under the action of vertical magnetic field (Li and Sun, J Differ Equ 267(6): 3827–3851, 2019). Under weak assumptions on the initial data which can be discontinuous, unbounded and large as well as involve transition to pure single-phase points or regions, we show existence of global weak solutions with finite energy. The essential novelty of this work, compared with previous works on the same model, is that transition to each single-phase flow is allowed without any constraints between adiabatic constants or two densities. It means that one of the phases can vanish in a point while the other can persist. The lack of enough regularity for each densities brings up essential difficulties in the two-component pressure compared with the single-phase model, i.e., compressible Navier-Stokes equations. The key points to achieve the main result rely on the variables reduction technique for the pressure function, domain separation, and some new estimates. As a byproduct, we obtain the existence of global weak solutions to the compressible MHD system without resistivity in two dimensions under the action of non-negatively vertical magnetic field, which represents a step forward to the study of the global large solution to the compressible MHD system without resistivity.

我们考虑多维可压缩二流体模型的狄利克雷问题。它由每种流体的连续性方程和混合物的动量方程组成。该模型可以从具有相等速度的可压缩双流体模型（Bresch 等人，在 Arch Rational Mech Anal 196:599–629, 2010 中）和 Vlasov-Fokker-Planck/可压缩 Navier- Stokes（Mellet 和 Vasseur，Commun Math Phys 281(3):573–596, 2008）（另见具有应力扩散的可压缩 Oldroyd-B 模型（Barrett 等人，Commun Math Sci 15:1265–1323, 2017）。另一个有趣的联系是它形式上是垂直磁场作用下二维无电阻率的可压缩磁流体动力学 (MHD) 流动方程 (Li and Sun, J Differ Equ 267(6): 3827–3851, 2019)。在对初始数据可能是不连续的、无界的和大的以及涉及到纯单相点或区域的过渡的弱假设下，我们证明了具有有限能量的全局弱解的存在。与之前在同一模型上的工作相比，这项工作的基本新颖之处在于，允许过渡到每个单相流，而绝热常数或两个密度之间没有任何限制。这意味着一个阶段可以在一个点上消失，而另一个可以持续存在。与单相模型（即可压缩 Navier-Stokes 方程）相比，每个密度缺乏足够的规律性给双分量压力带来了本质上的困难。实现主要结果的关键点依赖于压力函数、域分离、和一些新的估计。作为副产品，我们得到了在非负垂直磁场作用下二维无电阻率的可压缩MHD系统全局弱解的存在性，这代表着可压缩MHD全局大解的研究向前迈进了一步。没有电阻率的系统。