The Verigin problem with and without phase transition

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 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.

Well-Posedness and Stability Results for Some Periodic Muskat Problems

“…However, many of the mathematical studies on this topic are quite recent and they cover various physical scenarios and mathematical aspects related to the original model proposed in [52], cf. [6,7,9,[11][12][13][14][15][16][17][18][21][22][23][24][25]27,30,32,36,[38][39][40][41][42][43]48,49,49,53,54,58,[60][61][62] (see also [55,56] for some recent research on the compressible analogue of the Muskat problem, the so-called Verigin problem).…”

The Muskat problem with surface tension and equal viscosities in subcritical $$L_p$$-Sobolev spaces

“…Besides the fundamental well-posedness issue also other important features like the stability of stationary solutions [13,14,16,20,26,28,33,36], parabolic smoothing properties [24][25][26], the zero surface tension limit [8,31], and the degenerate limit when the thickness of the fluid layers (or a certain nondimensional parameter) vanishes [15,22,37] have been investigated in this context. We also refer to [19,21] for results on the Hele-Shaw problem with surface tension effects, which is the one-phase version of the Muskat problem, and to [34,35,38] for results on the related Verigin problem with surface tension.…”

The Muskat problem with surface tension and equal viscosities in subcritical $L_p$-Sobolev spaces

“…Besides the fundamental well-posedness issue also other important important features like the stability of stationary solutions [13,14,16,20,26,28,32,34], parabolic smoothing properties [24][25][26], the zero surface tension limit [8,31], and the degenerate limit when the thickness of the fluid layers (or a certain nondimensional parameter) vanishes [15,22] have been investigated in this context. We also refer to [19,21] for results on the Hele-Shaw problem with surface tension effects, which is the one-phase version of the Muskat problem, and to [35][36][37] for results on the related Verigin problem with surface tension.…”