Asymptotic justification of the conserved phase-field model with memory. (English) Zbl 0977.35139
Summary: We consider a conserved phase-field model with memory in which the Fourier heat conduction law is replaced by a constitutive assumption of Gurtin-Pipkin type; the system is conserved in the sense that the initial mass of the order parameter is preserved during the evolution. We investigate a Cauchy-Neumann problem for this model which couples an integro-differential equation with a nonlinear fourth-order equation for the phase field. Here we assume that the heat flux memory kernel has a decreasing exponential as principal part, and we study the behaviour of solutions when this kernel converges to a Dirac mass. We show that the solution to the conserved phase-field model with memory converges to a solution to the phase-field problem without memory under suitable assumptions on the data.
MSC:
| 35Q72 | Other PDE from mechanics (MSC2000) |
| 45K05 | Integro-partial differential equations |
| 35R35 | Free boundary problems for PDEs |
| 80A22 | Stefan problems, phase changes, etc. |
Keywords:
phase-field models; phase transitions; heat conduction with memory; asymptotic analysis; error estimates; heat flux memory kernel; Cauchy-Neumann problem; integro-differential equation; nonlinear fourth-order equationReferences:
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