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Flow properties for Young-measure solutions of semilinear hyperbolic problems

Published online by Cambridge University Press:  14 November 2011

Alexander Mielke
Alexander Mielke
Affiliation:
Institut für Angewandte Mathematik, Universität Hannover, Welfengarten 1, D-30167 Hannover ([email protected])
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Extract

For hyperbolic systems in one spatial dimension ∂tu + C∂xu = f(u), u(t, x) ∈ ℝd, we study sequences of oscillating solutions by their Young-measure limit, μ, and develop tools to study the evolution of μ directly from the Young measure, v, of the initial data. For d ≤ 2 we construct a flow mapping, St, such that μ(t) = St(v) is the unique Young-measure solution for initial value v. For d ≥ 3 we establish existence and uniqueness of Young measures that have product structure, that is the oscillations in direction of the Riemann invariants are independent. Counterexamples show that neither μ nor the marginal measures of the Riemann invariants are uniquely determined from v, except if a certain structural interaction condition for f is satisfied. We rely on ideas of transport theory and make use of the Wasserstein distance on the space of probability measures.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 1 , 1999 , pp. 85 - 123
Copyright
Copyright © Royal Society of Edinburgh 1999

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