In this chapter we will deal, on one hand, with the
study of the existence, uniqueness, regularity and estimates of
the solution of phase-field transition system, and on the other
hand, we will analyze the convergence of some approximating
schemes of fractional steps type, associated to this nonlinear
system.
system.
In the first Section the existence, uniqueness and regularity
of the solution of the phase-field transition system subject
to the non-homogeneous Cauchy-Neumann boundary conditions
(Theorem 2.1) are studied. Such kind of conditions
allow the phase-field system (Caginalp’s model) to be considered
as a model of heat transfer from the surface of product
to the environment, when we assume that heat is extracted by
convection and conduction in the continuous casting process.....
Keywords: nonlinear parabolic systems, abstract approximation
theory, nonlinear functional analysis, distributions, operator
theory, dissipative operators, degree theory, semigroups of
nonlinear operators, numerical analysis, fractional steps
method (Lie-Trotter product formula), stability and convergence
of numerical methods.
