Günther Grün

On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions

By means of energy and entropy estimates, we prove existence and positivity results in higher space dimensions for degenerate parabolic equations of fourth order with nonnegative initial values. We discuss their asymptotic behavior for

THERMODYNAMICALLY CONSISTENT, FRAME INDIFFERENT DIFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE TWO-PHASE FLOWS WITH DIFFERENT DENSITIES

t\to \infty

Nonnegativity preserving convergent schemes for the thin film equation

and give a counterexample to uniqueness.

Droplet Spreading Under Weak Slippage—Existence for the Cauchy Problem

A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is frame indifferent. Moreover, it is generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.

The thin-film equation: recent advances and some new perspectives

Summary. We present numerical schemes for fourth order degenerate parabolic equations that arise e.g. in lubrication theory for time evolution of thin films of viscous fluids. We prove convergence and nonnegativity results in arbitrary space dimensions. A proper choice of the discrete mobility enables us to establish discrete counterparts of the essential integral estimates known from the continuous setting. Hence, the numerical cost in each time step reduces to the solution of a linear system involving a sparse matrix. Furthermore, by introducing a time step control that makes use of an explicit formula for the normal velocity of the free boundary we keep the numerical cost for tracing the free boundary low.

Simulation of singularities and instabilities arising in thin film flow

Abstract In this paper, we consider the thin film equation u t + div(|u| n ∇Δu) = 0 in the multi-dimensional setting and solve the Cauchy problem in the parameter regime n ∈ [2, 3). New interpolation inequalities applied to the energy estimate enable us to control third order derivatives of appropriate powers of solutions. In such a way, a natural solution concept – reminiscent of that one used by Bernis and Friedman [Bernis, F., Friedman, A., (1990). Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83:179–206] in space dimension N = 1 − becomes available for the first time in the multi-dimensional setting. In addition, we provide the key integral estimate to establish results on the qualitative behavior of solutions like finite speed of propagation or occurrence of a waiting time phenomenon.

On a phase-field model for electrowetting

This paper is concerned with mathematical aspects of lubrication equations. In the first part, we discuss recent analytical achievements for various types of thin-film equations. Of interest are issues like (non-)uniqueness, wetting behaviour and contact line motion, in particular optimal propagation rates and waiting time or dead core phenomena. In the second part, we shall present novel numerical results for thin-film flow on heterogeneous substrates based on entropy consistent schemes. Finally, we will be concerned with new algorithmic concepts for the simulation of thin-film flow of shear-thinning liquids.

Lower bounds on waiting times for degenerate parabolic equations and systems

We present a nite element scheme for nonlinear fourth-order diusion equations that arise for example in lubrication theory for the time evolution of thin lms of viscous fluids. The equations are in general fourth-order degenerate parabolic, but in addition singular terms of second order may occur which model the eects of intermolecular forces or thermocapillarity. Discretizing the arising nonlinearities in a subtle way allows us to establish discrete counterparts of the essential integral estimates found in the continuous setting. As a consequence, the algorithm is ecient, and results on convergence, nonnegativity or even strict positivity of discrete solutions follow in a natural way. Applying this scheme to the numerical simulation of dierent models shows various interesting qualitative eects, which turn out to be in good agreement with physical experiments.

Droplet spreading under weak slippage: A basic result on finite speed of propagation

The term electrowetting is commonly used for phenomena where shape and wetting behavior of liquid droplets are changed by the application of electric fields. We develop and analyze a model for electrowetting that combines the Navier–Stokes system for fluid flow, a phase-field model of Cahn–Hilliard type for the movement of the interface, a charge transport equation, and the potential equation of electrostatics. The model is derived with the help of a variational principle due to Onsager and conservation laws. A modification of the model with the Stokes system instead of the Navier– Stokes system is also presented. The existence of weak solutions is proved for several cases in two and three space dimensions, either with non-degenerate or with degenerate electric conductivity vanishing in the droplet exterior. Some numerical examples in two space dimensions illustrate the applicability of the model. 2000 Mathematics Subject Classification: 35D05, 35D10, 35R35, 76M30.

Two-phase flow with mass density contrast

We extend the method in [19] to obtain quantitative estimates of waiting times for free boundary problems associated with degenerate parabolic equations and systems. Our approach is multidimensional, it applies to a large class of equations, including thin-film equations, (doubly) degenerate equations of second and of higher order and also systems of semiconductor equations. For these equations, we obtain lower bounds on waiting times which we expect to be optimal in terms of scaling. This assertion is true for the porous-medium equation which seems to be the only PDE for which two-sided quantitative estimates of the waiting time have been established so far.