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-a(2)partial derivative(x)(2))partial derivative(t)u + partial derivative(x) [b partial derivative(x)(2)u + f(u)] = 0, a > 0, b is an element of R, f is an element of C-Lip(R) with u : R-2 -> R. If we model the initial microstructure by a sequence of spatially faster and faster oscillating classical initial data nu(n), we obtain a sequence of spatially highly oscillatory classical solutions u(n). By considering the Young measures (YMs) nu and mu generated by the sequences nu(n) and u(n), respectively, as n -> infinity, we derive a macroscopic evolution equation for the YM solution mu, 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 nu(n) and non-linearities Copyright (c) 2005 John Wiley w Sons, Ltd.