On thermomechanical laws for the motion of a phase interface

This paper discusses the formulation of balance laws for mass, force, and energy in conjunction with a law of entropy growth for the motion of a sharp evolving phase interface within a continuum framework.

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Zusammenfassung In dieser Arbeit werden Erhaltungssätze für Masse, Kraft und Energie sowie eine Entropieungleichung für die räumliche Ausbreitung einer scharfen Phasengrenzschicht im Rahmen eines kontinuierlichen Modells diskutiert.

     

On a limiting motion and self-intersections for the intermediate surface diffusion flow

We rigorously prove that the solution surface of the intermediate surface diffusion flow converges to that of the averaged mean curvature flow locally in time as the diffusion coefficient tends to infinity. As an application of this convergence result, we show that the intermediate surface diffusion flow can drive embedded hypersurfaces into self-intersections. RID="*" ID="*"Partially supported by the Japan Society for the Promotion of Science, Grant No. 10304010, 12814024.     

On global motion of a compressible viscous heat-conducting fluid bounded by a free surface

We consider the motion of a viscous compressible heat-conducting fluid in ³ bounded by a free surface which is under constant exterior pressure. We present the global existence theorems in two cases: when the free surface is under the surface tension and without it.     

Convergence of diffusion generated motion to motion by mean curvature

We provide a new proof of convergence to motion by mean curvature (MMC) for the Merriman–Bence–Osher thresholding algorithm. The proof is elementary and does not rely on maximum principle for the scheme. The strategy is to construct a natural ansatz of the solution and then estimate the error. The proof thus also provides a convergence rate. Only some weak integrability assumptions of the heat kernel, but not its positivity, is used. Currently the result is proved in the case when smooth and classical solution of MMC exists.     

Problem of the motion of two compressible fluids separated by a closed free interface

We consider the problem of the simultaneous evolution for two barotropic capillary viscous compressible fluids occupying the space ℝ³ and separated by a closed free interface. Under some restrictions on the viscosities of the liquids, the local (in time) unique solvability of this problem is obtained in the Sobolev-Slobodetskii spaces. After the passage to Lagrangian coordinates it is possible to exclude the fluid density from the system of equations. The proof of the existence theorem for a nonlinear, noncoercive initial boundary-value problem is based on the method of successive approximations and on an explicit solution of a model linear problem with a plane interface between the liquids. The restrictions on the viscosities mentioned above appear in the intermediate estimation of this explicit solution in the Sobolev spaces with an exponential weight. Bibliography: 8 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 000, 1997, pp. 61–86. Translated by I. V. Denisova.     

Problem of the motion of two viscous incompressible fluids separated by a closed free interface

A local existence theorem for the problem of unsteady motion of a drop in a viscous incompressible capillary fluid is proved in Sobolev spaces. A linearized problem with known closed interface is also studied in Holder spaces of functions.     

On the slowness of phase boundary motion in one space dimension

We study the limiting behavior of the solution of with a Neumann boundary condition or an appropriate Dirichlet condition. The analysis is based on “energy methods”. We assume that the initial data has a “transition layer structure”, i.e., u^? ≈ ±+M 1 except near finitely many transition points. We show that, in the limit as ? → 0, the solution maintains its transition layer structure, and the transition points move slower than any power of ?.     

Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid

We consider a fully practical finite element approximation of the degenerate Cahn-Hilliard equation with elasticity: Find the conserved order parameter, , and the displacement field, , such that

subject to an initial condition on and boundary conditions on both equations. Here is the interfacial parameter, is a non-smooth double well potential, is the symmetric strain tensor, is the possibly anisotropic elasticity tensor, with and is the degenerate diffusional mobility. In addition to showing stability bounds for our approximation, we prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system in two space dimensions. Finally, some numerical experiments are presented.

     

Dopant diffusion in phases with a moving interface

We consider the diffusion of a dopant through a moving interface in the suicide film-Si system during silicide layer growth. The dopant concentration distribution is derived in analytical form by the integral Fourier transform method with subsequent reduction of the dopant redistribution problem to numerical solution of two integral equations. The results are presented in the form of curves plotting the time dependence of dopant concentration on both sides of the interface for various values of diffusion coefficients and interface velocity. The effect of physical parameters on the variation of dopant concentration near the interface is demonstrated.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 93–97, 1987.     

Unsteady motion of a finite mass of fluid,bounded by a free surface

It is proved that the problem with a free boundary for the Navier-Stokes equations, describing the motion of a finite mass of viscous, incompressible capillary fluid, has a unique solution for all t > 0 if the domain occupied by the fluid is nearly a ball and the velocity vector field is small at the initial moment.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 137–157, 1986.     

Diffraction of surface waves on a vertical interface

A problem of diffraction of waves on a vertical interface of media is studied. The waves are radiated by a point source. The reflected and refracted wave fields are obtained by the method of parabolic equation. The corresponding transformation coefficients are found for the Neumann boundary condition. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 70–89.     

On a multi-threshold compound Poisson process perturbed by diffusion

We consider a multi-layer compound Poisson surplus process perturbed by diffusion and examine the behaviour of the Gerber–Shiu discounted penalty function. We derive the general solution to a certain second order integro-differential equation. This permits us to provide explicit expressions for the Gerber–Shiu function depending on the current surplus level. The advantage of our proposed approach is that if the diffusion term converges to zero, the above-mentioned explicit expressions converge to those under the classical compound Poisson model, provided that the same initial conditions apply. This is subsequently illustrated by an extended example related to the probability of ultimate ruin.     

On diffusion of a single‐phase,slightly compressible fluid through a randomly fissured medium

In this paper, the Douglas–Peszyńska–Showalter model of diffusion through a partially fissured medium is given a stochastic formulation using the framework for problems in random media as set forth by Jikov, Kozlov and Oleinik. The concept of stochastic two‐scale convergence in the mean is then used to homogenize the randomized micromodels which result. As a consequence of this homogenization procedure, exact stochastic generalizations of results obtained by Clark and Showalter on diffusion through periodically fissured media are derived. Copyright © 2001 John Wiley & Sons, Ltd.     

Tension of a compound piezoceramic plate with a hole intersecting the phase interface

We study the two-dimensional electroelastic problem for an unbounded compound plate with an arbitrary hole located in both components of the plate. The corresponding boundary-value problem is reduced to a system of singular integral equations of second kind, which for the case of an elliptic hole can be solved numerically by the method of quadratures. We give the data of computations that characterize the concentration of the electroelastic fields near the hole subject to action at infinity by fields of mechanical stresses and electric tension. It is noted that in the case of the inverse piezoelectric effect the influence of inhomogeneities of the plate on the stress concentrations is sharply expressed. Six figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 67–75.