Cited By
View all- Otto FRump TSlepcev D(2006)Coarsening Rates for a Droplet ModelSIAM Journal on Mathematical Analysis10.1137/05063019238:2(503-529)Online publication date: 1-Jan-2006
Coarsening Rates in Off-Critical Mixtures
Pages 1732 - 1741
Abstract
We study coarsening of a binary mixture within the Mullins--Sekerka evolution in the regime where one phase has small volume fraction $\phi \ll1$. Heuristic arguments suggest that the energy density, which represents the inverse of a typical length scale, decreases as $\phi t^{-1/3}$ as $t \to \infty$. We prove rigorously a corresponding weak lower bound. Moreover, we establish a stronger result for the two-dimensional case, where we find a lower bound of the form $ \phi(\ln \phi^{-1})^{1/3}t^{-1/3}$. Our approach follows closely the analysis in [R. V. Kohn and F. Otto, Comm. Math. Phys., 229 (2002), pp. 375-395], which exploits the relation between two suitable length scales. Our main contribution is an isoperimetric inequality in the two-dimensional case.
References
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Nicholas Alikakos, Giorgio Fusco, Georgia Karali, Ostwald ripening in two dimensions—the rigorous derivation of the equations from the Mullins‐Sekerka dynamics, J. Differential Equations, 205 (2004), 1–49
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Shibin Dai, Robert Pego, Universal bounds on coarsening rates for mean‐field models of phase transitions, SIAM J. Math. Anal., 37 (2005), 347–371
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Robert Kohn, Felix Otto, Upper bounds on coarsening rates, Comm. Math. Phys., 229 (2002), 375–395
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Robert Kohn, Xiaodong Yan, Coarsening rates for models of multicomponent phase separation, Interfaces Free Bound., 6 (2004), 135–149
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Published: 01 January 2006
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View all- Otto FRump TSlepcev D(2006)Coarsening Rates for a Droplet ModelSIAM Journal on Mathematical Analysis10.1137/05063019238:2(503-529)Online publication date: 1-Jan-2006
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