The Verigin problem with and without phase transition

Jan Prüss¹ ; Gieri Simonett²

¹Martin-Luther-Universität Halle-Wittenberg, Germany
²Vanderbilt University, Nashville, USA

Interfaces and free boundaries, Tome 20 (2018) no. 1, pp. 107-128

Résumé

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 Lp​-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.

Détail

DOI : 10.4171/ifb/398

Classification : 35-XX, 76-XX
Mots-clés : Two-phase flows, phase transition, Darcy’s law, Forchheimer’s law, available energy, quasilinear parabolic evolution equations, maximal regularity, generalized principle of linearized stability, convergence to equilibria