In this paper we consider the evolution of two fluid phases in a porous medium. The fluids are separated from each other and also the wetting phase from air by interfaces which evolve in time. We reduce the problem to an abstract evolution equation. A generalised Rayleigh-Taylor condition characterizes the parabolicity regime of the problem and allows us to establish a general well-posedness result and to study stability properties of flat steady-states. When considering surface tension effects at the interface between the fluids and if the more dense fluid lies above, we find bifurcating finger-shaped equilibria which are all unstable.2010 Mathematics Subject Classification. 35B35; 35B36; 35K55; 35R37.
We consider the evolution of two thin fluid films in a porous medium. Starting from the classical equations modelling the Muskat problem we pass to the limit of small layer thickness and obtain a system of two coupled and degenerate parabolic equations for the films height. In the absence of surface tension forces we prove local well-posedness of the problem and show that the steady-states are exponentially stable.Mathematics Subject Classification (2010). Primary 76A20, 35Q35; Secondary 35B35, 35K51.
We present an explicit solution for the geophysical edge wave problem in the f-plane approximation. We also discuss in detail the properties of this solution and compare them with those of the solution found in the absence of Coriolis effects.
We study a moving boundary problem describing the growth of nonnecrotic tumors in different regimes of vascularisation. This model consists of two decoupled Dirichlet problem, one for the rate at which nutrient is added to the tumor domain and one for the pressure inside the tumor. These variables are coupled by a relation which describes the dynamic of the boundary. By re-expressing the problem as an abstract evolution equation, we prove local well-posedness in the small Hölder spaces context. Further on, we use the principle of linearised stability to characterise the stability properties of the unique radially symmetric equilibrium of the problem.2010 Mathematics Subject Classification. 35B35, 35B40, 35K55.
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