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 equation (1) with u: ?² → ?. If we model the initial microstructure by a sequence of spatially faster and faster oscillating classical initial data vⁿ, we obtain a sequence of spatially highly oscillatory classical solutions uⁿ. By considering the Young measures (YMs) ν and µ generated by the sequences vⁿ and uⁿ, 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 vⁿ and non‐linearities f. Copyright © 2005 John Wiley & Sons, Ltd.