Title: On the convergence to the smooth self-similar solution in the LSW model

Authors: Barbara Niethammer and Juan Luis Velazquez

Issue: Volume 55 (2006), Issue 2, 761-794

Abstract: We investigate the long-time behavior of solutions to the classical mean-field model by Lifshitz-Slyozov and Wagner (LSW). In the original work (see I.M. Lifshitz and V.V. Slyozov, \emph{The kinetics of precipitation from supersaturated solid solutions}, J. Phys. Chem. Solids \textbf{19} (1961), 35--50; C. Wagner, \emph{Theorie der Alterung von Niederschlaegen durch Umloesen}, Z. Elektrochemie \textbf{65} (1961), 581--594) convergence of solutions to a uniquely determined self-similar solution was predicted. However, it is by now well known (see B. Giron, B. Meerson, P.V. Sasorov, \emph{Weak selection and stability of localized distributions in Ostwald ripening}, Phys. Rev. E \textbf{58} (1998), 4213--4216; B. Niethammer, Robert L. Pego, \emph{Non-self-similar behavior in the LSW theory of Ostwald ripening}, J. Statist. Phys. \textbf{95} (1999), 867--902; B. Niethammer, Robert L. Pego, \emph{On the initial-value problem in the Lifshitz-Slyozov-Wagner theory of Ostwald ripening}, SIAM J. Math. Anal. \textbf{31} (2000), 467--485) that the long-time behavior of solutions depends sensitively on the initial data. In B. Niethammer, Robert L. Pego, \emph{Non-self-similar behavior in the LSW theory of Ostwald ripening}, J. Statist. Phys. \textbf{95} (1999), 867--902; B. Niethammer, Robert L. Pego, \emph{On the initial-value problem in the Lifshitz-Slyozov-Wagner theory of Ostwald ripening}, SIAM J. Math. Anal. \textbf{31} (2000), 467--485; B. Niethammer, J.J.L. Vel\'{a}zquez, \emph{Global stability and bounds for coarsening rates within the LSW mean-field theory}, Comm. Partial Differential Equations (2006) (to appear), a necessary and sufficient criterion for convergence to any self-similar solution which behaves like a finite power at the end of its (compact) support is given. In this paper we establish corresponding results for the LSW-solution which decays faster than any power. It turns out that the respective criterion for convergence to self-similarity is much less stringent than for the case of non-smooth self-similar solutions.