Stephan Luckhaus

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

The Gibbs-Thompson relation within the gradient theory of phase transitions

This paper discusses the asymptotic behavior as ɛ → 0+ of the chemical potentials λɛ associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid with free energy, per unit volume, given by ɛ2¦▽ϱ¦2+ W(ϱ), where ϱ is the density. The main result is that λɛ is asymptotically equal to ɛEλ/d+o(ɛ), with E the interfacial energy, per unit surface area, of the interface between phases, λ the (constant) sum of principal curvatures of the interface, and d the density jump across the interface. This result is in agreement with a formula conjectured by M. Gurtin and corresponds to the Gibbs-Thompson relation for surface tension, proved by G. Caginalp within the context of the phase field model of free boundaries arising from phase transitions.

Implicit time discretization for the mean curvature flow equation

In this paper we apply the method of implicit time discretization to the mean curvature flow equation including outer forces. In the framework ofBV-functions we construct discrete solutions iteratively by minimizing a suitable energy-functional in each time step. Employing geometric and variational arguments we show an energy estimate which assures compactness of the discrete solutions. An additional convergence condition excludes a loss of area in the limit. Thus existence of solutions to the continuous problem can be derived. We append a brief discussion of the related Mullins-Sekerka equation.

Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow

In this paper, we justify by periodic homogenization the double-porosity model for immiscible incompressible, two-phase flow. The volume fraction of the fissured part and the nonfissured part are kept positive constants and of the same order. The scaling is such that, in the final homogenized equations, the less permeable part of the matrix contributes as a nonlinear memory term. To prove the convergence of the total velocity and of the “reduced” pressure, we use the two-scale convergence since it seems to be appropriate for the problem, even though it would be possible to work with periodic modulation. However, in the final step, the degenerate ellipticity prevents the use of the two-scale convergence method and leads us to use periodic modulation.

Solutions for the Two-Phase Stefan Problem with the Gibbs—Thomson Law for the Melting Temperature

The coupling of the Stefan equation for the heat flow with the Gibbs-Thomson law relating the melting temperature to the mean curvature of the phase interface is considered. Solutions, global in time, are constructed which satisfy the natural a priori estimates. Mathematically the main difficulty is to prove a certain regularity in time for the temperature and the indicator function of the phase separately. A capacity type estimate is used to give an Lx bound for fractional time derivatives.

On nonstationary flow through porous media

SummarySaturated-unsaturated flow of an incompressible fluid through a porous medium is considered in the case of time-dependent water levels. This corresponds to coupling the mass conservation law with a continuous constitutive condition between water content and pressure. An existence result for the corresponding weak formulation is proved. Finally we study the limit as the constitutive relation degenerates into a maximal monotone graph.

Flow of Oil and Water in a Porous Medium

Abstract The flow of two immiscible and incompressible fluids in a porous medium is described by a system of quasilinear degenerate partial differential equations. In this paper the existence of a weak solution by regularization is shown.

Dynamical evolution of elasto-perfectly plastic bodies

We prove the existence of a displacement field and of a stress field that satisfy the dynamical equation for continuous media and the Prandtl-Reuss constitutive law of elasto-perfect plasticity. First we obtain the existence of a displacement rate in a space of functions of bounded deformation, where the constitutive law is satisfied in an integral form, then we show that one can choose a good representative for the stress in such a way that the Prandtl-Reuss law is satisfied almost everywhere with respect to the deformation measure.

Gradient estimates for solutions of parabolic equations and systems

1. STATEMENT OF THE PROBLEM Let 52 c R”, n > 2, be a bounded domain with smooth boundary r= &2. In this note we study scalar quasilinear parabolic differential equations u, = div,Y(g( IVul’) VU) +f(u) = 0, (1) u,-A,u+f(u)=O (2) and systems u, DA,u + f(u) = 0 (3) on Q x (0, co), with initial conditions u( ., 0) = u0 resp. u( ., 0) = u0 and homogeneous Dirichlet boundary conditions u,i-X(O,CO)‘O for (1) and (2) resp. (4) u1r.x (0.00) = 0 for (3). (5)

Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients

We consider a so-called random obstacle model for the motion of a hypersurface through a field of random obstacles, driven by a constant driving field. The resulting semi-linear parabolic PDE with random coefficients does not admit a global nonnegative stationary solution, which implies that an interface that was flat originally cannot get stationary. The absence of global stationary solutions is shown by proving lower bounds on the growth of stationary solutions on large domains with Dirichlet boundary conditions. Difficulties arise because the random lower order part of the equation cannot be bounded uniformly.