Isothermal compressible two-phase flows with and without phase transition are modeled, employing Darcy's and/or Forchheimer's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria are identified and their thermodynamical stability is investigated by means of a variational approach. It is shown that the problems are well-posed in an $L_p$-setting and generate local semiflows in the proper state manifolds. It is further shown that a non-degenerate equilibrium is dynamically stable in the natural state manifold if and only if it is thermodynamically stable. Finally, it is shown that a solution which does not develop singularities exists globally and converges to an equilibrium in the state manifold.

中文翻译：

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有和没有相变的 Verigin 问题

对有和没有相变的等温可压缩两相流进行建模，对速度场采用 Darcy 定律和/或 Forchheimer 定律。结果表明，在可用能量是严格的李雅普诺夫泛函的意义上，所得系统在热力学上是一致的。在这两种情况下，都确定了平衡，并通过变分方法研究了它们的热力学稳定性。结果表明，这些问题在 $L_p$ 设置中是适定的，并在适当的状态流形中生成局部半流。进一步表明，非简并平衡在自然状态流形中是动态稳定的，当且仅当它是热力学稳定的。最后，