The Verigin problem with and without phase transition

TL;DR

This paper models isothermal two-phase flows with and without phase transition, analyzing their thermodynamic stability, well-posedness, and long-term behavior, demonstrating the connection between thermodynamic and dynamic stability.

Contribution

It provides a comprehensive mathematical framework for analyzing the stability and dynamics of two-phase flows with phase transition, including well-posedness and convergence results.

Findings

Equilibria are identified and their thermodynamical stability is characterized.

The systems are shown to be well-posed in an $L_p$-setting and generate local semiflows.

Non-degenerate equilibria are dynamically stable if and only if thermodynamically stable.

Abstract

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.