Abstract. In this paper we develop a geometric theory for quasilinear parabolic problems in weighted Lp-spaces. We prove existence and uniqueness of solutions as well as the continuous dependence on the initial data. Moreover, we make use of a regularization effect for quasilinear parabolic equations to study the ω-limit sets and the long-time behaviour of the solutions. These techniques are applied to a free boundary value problem. The results in this paper are mainly based on maximal regularity tools in (weighted) Lp-spaces.

Abstract. We study the regularity of the free boundary arising in a thermodynamically consistent two-phase Stefan problem with surface tension by means of a family of parameter-dependent diffeomorphisms, Lp-maximal regularity theory, and the implicit function theorem.

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This article presents direct numerical simulations of single air bubbles and bubble pairs in water (with log Mo = −10.6) with a highly parallelized code based on the Volume Of Fluid method (VOF). Systematical simulations of terminal velocity of single bubbles with a diameter ranging from 0.5–15 mm (ReB = 200–3750) show good agreement with experimental data from Clift et al. Bubbles with a diameter of 8 mm show strong realistic surface deformations. Initial white noise has been added to all simulations to create realistic starting conditions. Rise paths of the bubbles depend strongly on the boundary conditions and the wall distance. Small wall distances reduce the path radii of the bubbles leading to an increased wake shedding frequency. For bubble pairs with wobbling surfaces the phenomenon of shedding of vortices from the edges of the bubbles is observed.

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.