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Niethammer, Barbara; Pego, Robert L.

doi:10.1023/a:1010405812125

Cited by 26 publications

(11 citation statements)

References 19 publications

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“…(3) and (4), for initial DFs with compact support and finite λ, different values of the scaled coefficient σ are obtained. Finally, for those initial DFs with compact support that cannot be described by a power-law asymptotics near R = R m , convergence to any self-similar solution is impossible [11]. A rigorous mathematical proof of these results is presently available [11].…”

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confidence: 99%

“…It has become clear after the analyses of Refs. [8,9,11] that in order to get strong selection, one must go beyond the "classical" model. In the present communication I report on a progress in this direction.…”

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confidence: 99%

“…Then, performing integration by parts in the two remaining terms, we again arrive at Eq. (11). In this derivation one should disregarded the boundary terms produced by integration by parts.…”

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confidence: 99%

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Fluctuations provide strong selection in Ostwald ripening

Meerson

1999

Phys. Rev. E

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A selection problem that appears in the Lifshitz-Slyozov (LS) theory of Ostwald ripening is reexamined. The problem concerns selection of a self-similar distribution function (DF) of the minority domains with respect to their sizes from a whole one-parameter family of solutions. A strong selection rule is found via an account of fluctuations. Fluctuations produce an infinite tail in the DF and drive the DF towards the "limiting solution" of LS or its analogs for other growth mechanisms.

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Fluctuations provide strong selection in Ostwald ripening

Meerson

1999

Phys. Rev. E

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“…Furthermore, the uniqueness of the pair (G, v) follows from [18] if G 0 is compactly supported, and from [19] in the general case. (Only the case α = 1/3 is actually considered in [18,19], but their proofs extend to α ∈ (0, 1).) Let us finally point out that the integrability assumption G 0 ∈ L 1 (R + ; xdx) is not needed for the existence of a solution to (2.1), (2.2), (2.3).…”

Section: Resultsmentioning

confidence: 99%

“…Before describing our results more precisely, let us recall that the LSW equation (1.1)-(1.2) has been the object of several studies recently; existence and uniqueness of weak solutions have been proved in [10,17,19] for the initial value problem (1.1)-(1.2) under various assumptions on the functions k and q determining the growth rate of the grains V and the initial data. Also, the large time asymptotics have been investigated in [1,16] by analytical means and in [2,6] …”

Section: Introductionmentioning

confidence: 99%

Numerical Approximation of the Lifshitz--Slyozov--Wagner Equation

Filbet

Laurençot

2003

SIAM J. Numer. Anal.

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The Lifshitz-Slyozov-Wagner theory of coarsening (Ostwald ripening) describes the late stages of the growth by diffusional mass transfer of the grains of a new phase from a supersaturated solution. It results in a nonlinear transport equation with a nonlocal nonlinearity for the volume distribution function of the grains. A time explicit finite volume numerical scheme is proposed to solve this equation in self-similar variables and is shown to converge under a CFL condition. Numerical simulations are also presented.

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Self–similarity and instability in classical mean–field models for domain coarsening

Niethammer¹,

Pego²

2002

Proc. Appl. Math. Mech.

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In the classical theory by Lifshitz, Slyozov and Wagner (LSW) coarsening of a dilute system of particles is modelled by a nonlocal transport equation for the particle size distribution. LSW predict that the asymptotic behavior for large times is self-similar and that a particular self-similar profile is approached. In this talk we discuss rigorous results on the long-time behavior of solutions for several variants of this model. For systems in which particle size is uniformly bounded these results establish a sensitive dependence on the data and thus in general do not confirm the predictions by LSW. More precisely we prove that convergence to the classically predicted similarity solution is impossible if the initial distribution is comparable to any finite power of distance to the end of the support. In addition we give a necessary criterion for convergence to other self-similar solutions, which implies non-self-similar asymptotics for a dense set of data.

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Cited by 26 publications

References 19 publications

Fluctuations provide strong selection in Ostwald ripening

Fluctuations provide strong selection in Ostwald ripening

Numerical Approximation of the Lifshitz--Slyozov--Wagner Equation

Self–similarity and instability in classical mean–field models for domain coarsening

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