Global Stability and Bounds for Coarsening Rates Within the Mean-Field Theory for Domain Coarsening

We investigate the long-time behavior of solutions to the classical mean-field model for coarsening by Lifshitz–Slyozov and Wagner (LSW). In the original work (Lifshitz and Slyozov, 1961 Lifshitz , I. M. , Slyozov , V. V. ( 1961 ). The kinetics of precipitation from supersaturated solid solutions . J. Phys. Chem. Solids 19 : 35 – 50 . CSA] CROSSREF] Crossref], Web of Science ®] , Google Scholar]; Wagner 1961 Wagner , C. ( 1961 ). Theorie der Alterung von Niederschlägen durch Umlösen . Z. Elektrochemie 65 : 581 – 594 . CSA]  Google Scholar]) convergence of solutions to a uniquely determined self-similar solution was predicted. However, it is by now well known (Giron et al., 1998 Giron , B. , Meerson , B. , Sasorov , V. P. ( 1998 ). Weak selection and stability of localized distributions in Ostwald ripening . Phys. Rev. E 58 : 4213 – 4216 . CSA] CROSSREF] Crossref], Web of Science ®] , Google Scholar]; Niethammer and Pego 1999 Niethammer , B. , Pego , R. L. ( 1999 ). Non-self-similar behavior in the LSW theory of Ostwald ripening . J. Stat. Phys. 95 ( 5/6 ): 867 – 902 . CSA] CROSSREF] Crossref], Web of Science ®] , Google Scholar] 2001 Niethammer , B. , Pego , R. L. ( 2001 ). The LSW model for domain coarsening: Asymptotic behavior for total conserved mass . J. Stat. Phys. 104 ( 5/6 ): 1113 – 1144 . CSA] CROSSREF] Crossref], Web of Science ®] , Google Scholar]) that the long-time behavior of solutions depends sensitively on the initial data. In Niethammer and Pego (1999 Niethammer , B. , Pego , R. L. ( 1999 ). Non-self-similar behavior in the LSW theory of Ostwald ripening . J. Stat. Phys. 95 ( 5/6 ): 867 – 902 . CSA] CROSSREF] Crossref], Web of Science ®] , Google Scholar] 2001 Niethammer , B. , Pego , R. L. ( 2001 ). The LSW model for domain coarsening: Asymptotic behavior for total conserved mass . J. Stat. Phys. 104 ( 5/6 ): 1113 – 1144 . CSA] CROSSREF] Crossref], Web of Science ®] , Google Scholar]) a necessary criterion for convergence to any self-similar solution which behaves like a finite power at the end of its (compact) support is given. It says that the data have to be regularly varying at the end of their support with the same power. This criterion is also shown to be sufficient if the power is sufficiently small and for data which are close to self-similar.

In this article we extend the local stability result to the whole range of self-similar solutions with compact support. Our first main result establishes global stability of self-similar solutions with not too large power. The proof relies on a global contraction argument for the spreading of characteristics. In addition, we also establish upper and lower bounds for the coarsening rates of the system for a suitable class of initial data whose variation is bounded at the end of the support but not necessarily regular.