The Rayleigh-Taylor instability for the verigin problem with and without phase transition. (English) Zbl 1418.35310
Summary: Isothermal compressible two-phase flows in a capillary are modeled with and without phase transition in the presence of gravity, employing Darcy’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 with flat interface are identified. It is shown that the problems are well-posed in an \(L_p\)-setting and generate local semiflows in the proper state manifolds. The main result concerns the stability of equilibria with flat interface, i.e. the Rayleigh-Taylor instability.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76D27 | Other free boundary flows; Hele-Shaw flows |
| 76E17 | Interfacial stability and instability in hydrodynamic stability |
| 35R37 | Moving boundary problems for PDEs |
| 35K59 | Quasilinear parabolic equations |
Keywords:
two-phase flows; phase transition; Darcy’s law with gravity; available energy; quasilinear parabolic evolution equations; maximal regularity; generalized principle of linearized stabilityReferences:
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