Merdan, H.; Caginalp, G. Renormalization and scaling methods for quasi-static interface problems. (English) Zbl 1153.82331 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 63, No. 5-7, 812-822 (2005). 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. Cited in 1 Document 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 Keywords:renormalization group; scaling; interface dynamics; quasi-static regime; capillarity length PDFBibTeX XMLCite \textit{H. Merdan} and \textit{G. Caginalp}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 63, No. 5--7, 812--822 (2005; Zbl 1153.82331) Full Text: DOI References: [1] Caginalp, G., Phys. Rev. E, 60, 6267 (1999) [2] Caginalp, G., SIAM J. Appl. Math., 62, 424 (2001) [3] Caginalp, G.; Merdan, H., Physica D, 198, 136 (2004) [4] Creswick, R. J.; Farach, H. A.; Poole, C. P., Introduction to Renormalization Group Methods in Physics (1992), Wiley: Wiley New York · Zbl 0771.60093 [5] Cross, M. C.; Hohenberg, P. C., Rev. Mod. Phys., 65, 851 (1993) [6] Goldenfeld, N., Lectures on Phase Transitions and the Renormalization Group (1992), Addison-Wesley: Addison-Wesley Reading, MA [7] Goldenfeld, N.; Martin, O.; Oono, Y.; Liu, F., Phys. Rev. Lett., 64, 1361 (1990) [8] Goldstein, R. E.; Pesci, A. I.; Shelley, M. J., Phys. Rev. Lett., 70, 3043 (1993) [9] Jasnow, D.; Vinals, J., Phys. Rev. A, 40, 3864 (1989) [10] Jasnow, D.; Vinals, J., Phys. Rev. A, 41, 6910 (1990) [11] Jasnow, D.; Yeung, C., Phys. Rev. E, 47, 1087 (1993) [12] Merdan, H.; Caginalp, G., Discrete Contin. Dyn. Syst. B, 3, 565 (2003) [13] Moise, I.; Temam, R., J. Dynam. Differential Equations, 13, 275 (2001) [14] Mullins, W. W.; Sekerka, R., J. Appl. Phys., 34, 323 (1963) [15] Mullins, W. W.; Sekerka, R., J. Appl. Phys., 35, 444 (1964) [16] J.R. Ockendon, Free Boundary Problems, vol. II, Ist. Naz. Alta Mat. Francesco Severi, Rome, 1980, p. 443.; J.R. Ockendon, Free Boundary Problems, vol. II, Ist. Naz. Alta Mat. Francesco Severi, Rome, 1980, p. 443. [17] Oleinik, O. A., Sov. Math. Dokl., 1, 1350 (1960) [18] Saffman, P. G.; Taylor, G. I., Proc. R. Soc. London Ser. A, 245, 312 (1958) [19] Zhang, Q.; Graham, M. J., Phys. Rev. Lett., 79, 2674 (1997) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.