In this paper, we propose a new viscosity for a coupled compressible Navier–Stokes/Allen–Cahn system because it describes the motion of a gas in a flowing liquid. The viscosity depends on two different variables (the density and the unknown function in Allen–Cahn equations). We establish blow-up criterions for strong solutions to initial-boundary value problem. We also show that the strong solutions can be improved to classical solutions. Precisely, when the viscosity only depends on the density, we prove the global existence and uniqueness of the Navier–Stokes/Allen–Cahn equations.

中文翻译：

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Navier–Stokes / Allen–Cahn系统的爆破判据和强/经典解的全球存在

在本文中，我们为可压缩的Navier–Stokes / Allen–Cahn耦合系统提出了新的粘度，因为它描述了气体在流动液体中的运动。粘度取决于两个不同的变量（Allen-Cahn方程的密度和未知函数）。我们为有效解决初边值问题建立了爆炸标准。我们还表明，强大的解决方案可以改进为经典解决方案。准确地说，当粘度仅取决于密度时，我们证明了Navier–Stokes / Allen–Cahn方程的整体存在性和唯一性。