Cooperation with: N. Kenmochi (Chiba University, Japan), P. Krejcí (Academy of Sciences of the Czech Republic, Prague), U. Stefanelli (Università di Pavia, Italy), C. Verdi (Università di Milano, Italy), S. Zheng (Fudan University, Shanghai, China)
Supported by: DFG: ``Hysterese-Operatoren in Phasenfeld-Gleichungen'' (Hysteresis operators in phase-field equations)
To be able to deal with phase transitions, one has to take into account diffusive effects as well as hysteretic phenomena.
The hysteretic phenomena are modeled by hysteresis operators.
A typical example is the
multi-dimensional stop operator with the characteristic set Z ,
which is defined for a convex, closed subset Z of .For a final time T > 0,
this operator maps a pair (u,x0) consisting of an
input function and some to the
solution to the variational inequality
where denotes the inner product on .The stop operator is used to define the corresponding multi-dimensional Prandtl-Ishlinskii operators
where is a real-valued function and, for ,is an initial value for the stop operator SrZ with the characteristic set rZ. In , these hysteresis operators are investigated, and for a phase-field system involving multi-dimensional Prandtl-Ishlinskii operators the existence of a unique solution is proved. This is an extension of results in the previous work (), since it is no longer required that the domain Z must be polyhedral. More general classes of multi-dimensional Prandtl-Ishlinskii operators have been studied in .
The mathematical modeling of nonlinear thermo-visco-plastic
developments leads to the following system
where u, , , and w are the unknowns displacement, absolute temperature, elastoplastic stress, and freezing index, respectively, , , CV, and are positive constants, f, g are given functions, and , and are hysteresis operators. In , this system has been derived, the thermodynamic consistency of the model has been proved, and the existence of a unique strong solution to an initial-boundary value problem for this system has been shown. A large time asymptotic result for this system is presented in . For almost the same problem with a more general boundary condition, similar existence and asymptotic results have been derived. An approach used in  to derive uniform estimates for the solutions to partial differential equations involving hysteresis operators has been further investigated in . Moreover, in , generalizations of the scalar Prandtl-Ishlinskii operators are introduced and investigated with respect to their thermodynamic consistency.
The system (3)-(6) can be simplified by adding uxxxx on the left-hand side of (3). The corresponding system has been considered in .
Phase-field systems of Penrose-Fife type are models for diffusive
phase-transition phenomena, see .
For a non-conserved scalar order parameter , one
gets the system
In this system CV, , and are given positive constants, and , , s, g are given functions.
In , an a posteriori error estimate has been derived for a time-discrete scheme for the system (7)-(8), with being a positive constant, and .
The system (7)-(8) with depending on can be used to model the anisotropic solidification of liquids. The existence of a solution to this system has been shown in .
Following [1, 2], the system (7)-(8) is modified by replacing in (8) by an integral operator and allowing to be a function of . For an integrodifferential (non-local) system derived in this way, results concerning global existence, uniqueness, and large-time asymptotic behavior have been derived in .
In another modification of the phase-field system,
the energy balance
(7) is coupled with two order parameter
where w is a scalar-valued order parameter, is an order parameter with values in ,and is the subdifferential of the indicator function IZ of a bounded, closed, convex set .Formally, this system can be considered as a diffusive approximation of the corresponding multi-dimensional stop operator .Results concerning existence, uniqueness, and continuous dependence on data have been presented in .
In , it has been proved for another phase-field system that the solutions are approximations for the evaluation of the stop operator.