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Renormalization and scaling methods for quasi-static interface problems. (English) Zbl 1153.82331

Summary: We study the temporal evolution of an interface separating two phases for its large-time behavior by adapting renormalization group methods and scaling theory. We consider a full two-phase model in the quasi-static regime and implement a renormalization procedure in order to calculate the characteristic length of a self-similar system, \(R(t)\), that is the time-dependent length scale characterizing the pattern growth. When the dynamical undercooling is non-zero (\(\alpha \neq 0\)), we find that \(R(t)\) increases as \(t^{-1/\lambda}\), where \(\lambda \) can take on values in the continuous spectrum, \([-3,-2]\). For \(\alpha =0\) the spectrum is \([-3,0)\) so that the single value of \(\lambda =-1\) is selected by the plane wave imposed by Jasnow and Vinals. It is also shown that in almost all of these cases, the capillarity length, \(d_{0}\), (arising from the surface tension, \(\sigma _{0}\)) is not relevant for the large-time behavior even though it has a crucial role at the early stage evolution of an interface. The exception is \(\lambda =-3\), i.e., \(R(t)\sim t^{1/3}\), for which \(d_{0}\) is invariant.

MSC:

82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
35K55 Nonlinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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