Hans Wilhelm Alt

Quasilinear elliptic-parabolic differential equations

The structure conditions are the ellipticity of a and the (weak) monotonicity of b, and b has to be a subgradient in case m > 1. First we treat the case that b is continuous, and later (Sect. 4) we include Stefan problems, that is, we allow b to have jumps. The special cases of an elliptic equation with time as parameter, that is, b(z)= 0, and the standard parabolic equation, that is, b(z)=z are included. Some special single equations of mixed elliptic and parabolic type are given in the following. The gas flow through a porous medium is described by the equation

A mathematical model of dynamics of non-isothermal phase separation

Abstract A mathematical model of thermally activated phase separation in binary systems is proposed. The construction is based on the Landau-Ginzburg theory of phase transitions and non-equilibrium thermodynamics. The proposed system of differential equations represents an extension of the Cahn-Hilliard model to the non-isothermal situation. The model is shown to be consistent with the first principles of thermodynamics. The stability solutions is considered, and for the one-dimensional case numerical results are presented.

Phase boundary dynamics: transitions between ordered and disordered lipid monolayers

Based on a general thermodynamical theory of mass and momentum, we propose and investigate a new phase field model for small transition layers between two spatially separated phases with intersecting free energy functions. We use a phase fraction that only depends on the ratio of the two density components. From the phase field model we derive conditions for the sharp interface velocity and density jumps. The general model is motivated by and applied to the dynamics of lipid monolayers, which appear as surfactant on the strongly expanded and compressed thin water film of lung alveoli. While the liquid condensed ordered phase (LC) of a flat lipid monolayer is characterized by high viscosity and limited compressibility, the liquid expanded disordered phase (LE) is dominated by diffusion and high compressibility. In order to perform the asymptotic transition layer analysis at moving phase boundaries, a new nonlinear free energy interpolation model is proposed whose excess energy, in comparison to standard linear interpolations, contains an energy hump that has to be surpassed in a permissive transition from one phase to the other. This leads to a unique density jump condition in the case that the ordered phase is extending, whereas in the retracting case the jump densities are not restricted. The transition profiles and the resulting interface speed are numerically determined for a typical example by solving a nonlinear degenerate ODE system. In a simplified 1-dimensional situation with low Reynolds number, the approximate macroscopic system of differential equations with moving sharp interface is numerically solved and interpreted in application to surfactant monolayers in lung alveoli.

A stationary flow of fresh and salt groundwater in a coastal aquifer

Consider a horizontally situated homogeneous aquifer of constant thickness h. In this aquifer, fresh and salt water are present and separated by an abrupt interface Γ. Let the flow be two dimensional in the vertical (x,z) plane

A free boundary problem involving a cusp. I : global analysis

We consider the behaviour of the interface (free boundary) between fresh and salt water in a porous medium (a reservoir). The salt water is below the interface (with respect to the direction of gravity) and is stagnant. The fresh water is above the interface and moves towards the wells which are present in the reservoir. We give a description of the corresponding flow problem leading to a weak variational formulation involving a parameter Q which is related to the strength of the wells. We show that Q is a critical parameter in the following sense: there exists Qcr > 0 such that for Q <Qcr a smooth interface exists which is monotone with respect to Q. For Q = Qcr, a free boundary with one or more singularities (cusps) will occur at a positive distance from the wells. The global analysis for the problem (existence, uniqueness, monotonicity) is given here for two and three dimensional flow situations. The local cusp analysis is two-dimensional, and will be discussed in Part II.

Dynamics of Non-Isothermal Phase Separation

A mathematical model of non-isothermal phase separation in binary systems is presented. The model, constructed within the Landau-Ginzburg theory of phase transitions, has the form of a coupled system of evolutionary nonlinear equations that describe mass diffusion and heat conduction in a quenched system. Existence of weak solutions to the model is discussed. Numerical results are presented in the case of one space dimension.

Distributional equation in the limit of phase transition for fluids

We study the convergence of a diffusive interface model to a sharp interface model. The model consists of the conservation of mass and momentum, where the mass undergoes a phase transition. The equations were considered in [W3] and in the diffuse case consist of the compressible Navier– Stokes system coupled with an Allen–Cahn equation. In the sharp interface limit a jump in the mass density as well as in the velocity occurs. The convergence of mass and momentum is considered in the distributional sense. The convergence of the free energy to a limit is shown in a separate paper. The procedure in this paper works also in other general situations.

A free boundary problem involving a cusp: breakthrough of salt water

textabstractIn this paper we study a two-phase free boundary problem describing the stationary flow of fresh and salt water in a porous medium, when both fluids are drawn into a well. For given discharges at the well (

Fluid Mixtures and Applications to Biological Systems

Q_f

Spectrum of compact operators

for fresh water and