Irena Pawłow

A generalized Stefan problem in several space variables

A multidimensional, multiphase problem of Stefan type, involving quasilinear parabolic equations and nonlinear boundary conditions is considered. Regularization techniques and monotonicity methods are exploited. Existence and uniqueness of a weak solution to the problem, as well as continuous and monotone dependence of the solution upon data are shown.

On a Class of Cahn--Hilliard Models with Nonlinear Diffusion

In the present work, we address a class of Cahn--Hilliard equations characterized by a nonlinear diffusive dynamics and possibly containing an additional sixth order term. This model describes the separation properties of oil-water mixtures when a substance enforcing the mixing of the phases (a surfactant) is added. However, the model is also closely connected with other Cahn--Hilliard-like equations relevant in different types of applications. We first discuss the existence of a weak solution to the sixth order model in the case when the configuration potential of the system has a singular (e.g., logarithmic) character. Then we study the behavior of the solutions in the case when the sixth order term is allowed to tend to 0, proving convergence to solutions of the fourth order system in a special case. The fourth order system is then investigated by a direct approach, and existence of a weak solution is shown under very general conditions by means of a fixed point argument. Finally, additional properties...

Parabolic-elliptic free boundary problems with time-dependent obstacles

In the paper, parabolic-elliptic problems with time-dependent obstacles and variable boundary conditions are considered. The obstacles are either concentrated at a given time-dependent part of the boundary of a geometric domain or within a prescribed time-dependent subdomain. The problems are fourmulated as a Cauchy problem in Hilbert space. Existence and uniqueness results are established, in particular refering to some models of flows in porous media and electrochemical machining processes with moving control actions.

QUASI-LINEAR THERMOELASTICITY SYSTEM ARISING IN SHAPE MEMORY MATERIALS ∗

In this paper we establish the global existence and uniqueness of a solution for the three‐dimensional and two‐dimensional forms of the quasi‐linear thermoelasticity system which arises as a mathematical model of shape memory alloys. The system represents a multidimensional version with viscosity and capillarity of the well‐known Falk model for one‐dimensional martensitic phase transitions. In the setup considered by Pawlow and Zaja¸czkowski [Math. Methods Appl. Sci., 28 (2005), pp. 407–442; 551–592], some conditions have been required for the nonlinear term. In the present paper we improve the result by imposing less restrictive assumptions.

Global Regular Solutions to a Kelvin--Voigt Type Thermoviscoelastic System

A classical three-dimensional thermoviscoelastic system of Kelvin--Voigt type is considered. The existence and uniqueness of a global regular solution is proved without small data assumption. The existence proof is based on the successive approximation method. The crucial part constitutes a priori estimates on an arbitrary finite time interval, which are derived with the help of the theory of anisotropic Sobolev spaces with a mixed norm.

Asymptotic behaviour of solutions to parabolic-elliptic variational inequalities

in a real Hilbert space H, where J is a given interval in R, = [0, co), &#J’ is the subdifferential of a convex function 4’ on H, and B is a maximal monotone operator in H. The abstract setting reflects the form of a class of parabolic-elliptic boundary value problems with time-dependent obstacles. Such problems arise, in particular, as mathematical models of various processes in electrochemical technology and some flows with saturations in porous media. Time-dependent obstacles may represent geometric controls (moving control actions) in these problems. In particular, such controls may be comprehended in variable boundary conditions prescribed over time-dependent parts of the boundary of domain or in source terms distributed over time-dependent subdomains. The former refers to control of flows via variable water table in reservoirs, the latter characterizes control of electrochemical process via mechanically driven motion of a toolpiece (cathode). The most typical control objective consists there in stabilizing the state of the process at a given spatial distribution (for example constant water head and desired shape of anode). This motivates our interest in the asymptotic behavior of solutions to the corresponding boundary value problems as t +a. To the knowledge of the authors, this question has so far not been studied in literature. As it has been shown in Kenmochi-Pawlow [16, 171, the class of considered time-dependent problems admits a representation in the form of evolution equation E(#, B). In [16, 171, the corresponding results on existence, uniqueness and stability of solutions have been established. In the present paper we study the asymptotic properties of solutions to E(c

Optimal Control of Two-Phase Stefan Problems — Numerical Solutions

‘, B) as t + 00, provided some hypotheses are imposed on the family (

Numerical solution of a multidimensional two-phase Stefan problem

‘; t E R,) and the limit 4” of # as t -+ a. For the solution (u, u*) of E(d’, B) on R, we prove that

Dynamics of Non-Isothermal Phase Separation

A method of constructing optimal controls for two-phase Stefan problems is proposed. The control action is performed via boundary conditions of the mixed type. An elliptic degeneration of the Stefan problems is admitted. Discrete approximations to the optimal control problems are constructed. A discussion of the results of some corresponding numerical experiments is given.

Transmission-Stefan Problems Arising in Czochralski Process of Crystal Growth

This paper treats a multidimensional two-phase Stefan problem with variable coefficients and mixed type boundary conditions. A numerical method for solving the problem is of fixed domain type, based on a variational inequality formulation of the problem. Numerical solutions are obtained by using piecewise linear finite elements in space and finite difference in time, and by solving a strictly convex minimization problem at each time step. Some computational results are presented.