Young-measure solutions to a generalized Benjamin–Bona–Mahony equation
✍ Scribed by Johannes Giannoulis
- Publisher
- John Wiley and Sons
- Year
- 2005
- Tongue
- English
- Weight
- 265 KB
- Volume
- 28
- Category
- Article
- ISSN
- 0170-4214
- DOI
- 10.1002/mma.587
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✦ Synopsis
Abstract
We consider the evolution of microstructure under the dynamics of the generalized Benjamin–Bona–Mahony equation
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with u: ℝ^2^ → ℝ. If we model the initial microstructure by a sequence of spatially faster and faster oscillating classical initial data v^n^, we obtain a sequence of spatially highly oscillatory classical solutions u^n^. By considering the Young measures (YMs) ν and µ generated by the sequences v^n^ and u^n^, respectively, as n → ∞, we derive a macroscopic evolution equation for the YM solution µ, and show exemplarily how such a measure‐valued equation can be exploited in order to obtain classical evolution equations for effective (macroscopic) quantities of the microstructure for suitable initial data v^n^ and non‐linearities f. Copyright © 2005 John Wiley & Sons, Ltd.