Flow properties for Young-measure solutions of semilinear hyperbolic problems

For hyperbolic systems in one spatial dimension partial derivative(t)u + C partial derivative(x)u = f(u), u(t, x) is an element of R-d, we study seque nces of oscillating solutions by their Young-measure limit, mu, and develop tools to study the evolution of mu directly from the Young measure, nu, of the initial data. For d less than or equal to 2 we construct a flow mappin g, S-t, such that mu(t) = S-t(nu) is the unique Young-measure solution for initial value nu. For d greater than or equal to 3 we establish existence a nd uniqueness of Young measures that have product structure, that is the os cillations in direction of the Riemann invariants are independent. Countere xamples show that neither mu nor the marginal measures of the Riemann invar iants are uniquely determined from nu, except if a certain structural inter action condition for f is satisfied. We rely on ideas of transport theory a nd make use of the Wasserstein distance on the space of probability measure s.