Flow properties for Young-measure solutions of semilinear hyperbolic systems

We consider a semilinear hyperbolic system of d equations of first order. We consider sequences of initial data which approximate a Young measure and study the Young-measure limit of the associated sequence of solutions. Thus we are able to define the notion of Cauchy problem and semigroups for Young measure solutions, at least for d≤2 by using compactness through compensation. For d≥3 the subclass of product measure solutions is considered, where each Cauchy problem has a unique solution. Continuous dependence of the associated semigroup on the initial data can be established in the Wasserstein topology and, under additional structural condition on the nonlinearity, also in the weak^* topology. The reported research extends previous work of L. Tartar and is partly joint work with Florian Theil.