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.