Young‐measure solutions to a generalized Benjamin–Bona–Mahony equation

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.