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Young‐measure solutions to a generalized Benjamin–Bona–Mahony equation
Author(s) -
Giannoulis Johannes
Publication year - 2005
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.587
Subject(s) - evolution equation , measure (data warehouse) , mathematics , sequence (biology) , microstructure , structural equation modeling , mathematical analysis , order (exchange) , statistics , computer science , materials science , database , biology , metallurgy , genetics , finance , economics
We consider the evolution of microstructure under the dynamics of the generalized Benjamin–Bona–Mahony equationwith 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.