Probabilistic estimates of the maximum norm of random Neumann Fourier series

•Problem arising in spinodal decomposition given by the stochastic Cahn–Hilliard model.•Study of extremal values of the pattern described by random cosine sums.•Rigorous asymptotic results as the wave number of pattern converges to infinity.•Numerical simulations of the maximum for medium range wave...

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Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation 2017-06, Vol.47, p.348-369
Main Authors: Blömker, Dirk, Wacker, Philipp, Wanner, Thomas
Format: Article
Language:English
Subjects:
Alloys
Boundary conditions
Computer simulation
Cosine series
Decoupling
Fourier analysis
Fourier series
Mathematical models
Maxima
Maximum norm
Modeling extremal values
Numerical experiments
Phase separation
Probability
Random Fourier series
Spinodal decomposition
Studies
Wavelengths
Citations: Items that this one cites
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Summary:•Problem arising in spinodal decomposition given by the stochastic Cahn–Hilliard model.•Study of extremal values of the pattern described by random cosine sums.•Rigorous asymptotic results as the wave number of pattern converges to infinity.•Numerical simulations of the maximum for medium range wave numbers.•Simplified model describing essential features of the magnitude of extremal values. We study the maximum norm behavior of L2-normalized random Fourier cosine series with a prescribed large wave number. Precise bounds of this type are an important technical tool in estimates for spinodal decomposition, the celebrated phase separation phenomenon in metal alloys. We derive rigorous asymptotic results as the wave number converges to infinity, and shed light on the behavior of the maximum norm for medium range wave numbers through numerical simulations. Finally, we develop a simplified model for describing the magnitude of extremal values of random Neumann Fourier series. The model describes key features of the development of maxima and can be used to predict them. This is achieved by decoupling magnitude and sign distribution, where the latter plays an important role for the study of the size of the maximum norm. Since we are considering series with Neumann boundary conditions, particular care has to be placed on understanding the behavior of the random sums at the boundary.
ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2016.11.023