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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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References

1Ash, R.. Information theory (Wiley, 1965).Google Scholar
2Ball, J. M.. A version of the fundamental theorem for Young measures. In Proc. of the CNRS–NSF Workshop on Continuum Theory of Phase Transitions (ed. Rascle, M., Serre, D. and Slemrod, M.). Springer Lecture Notes in Physics, no. 344, pp. 207–221 (Springer, 1989).Google Scholar
3Batt, J.. Ein Existenzbeweis für die Vlasov–Gleichung der Stellardynamik bei gemittelter Dichte. Arch. Ration. Mech. Analysis 13 (1963), 296–308.CrossRefGoogle Scholar
4Demoulini, S.. Young measure solutions for a nonlinear parabolic equation of forward–backward type. SIAM J. Math. Analysis 27 (1996), 376–403.CrossRefGoogle Scholar
5DiPerna, R. J.. Measure-valued solutions to conservation laws. Arch. Ration. Mech. Analysis 88 (1985), 223–270.CrossRefGoogle Scholar
6W. E, and Kohn, R. V.. The initial-value problem for measure-valued solutions of a canonical 2 × 2 system with linearly degenerate fields. Commun. Pure Appl. Math. 44 (1991), 981–1000.Google Scholar
7Fonseca, I., Brandon, D. and Swart, P.. Dynamics and oscillatory microstructure in a model of displacive phase transformations. In Progress in partial differential equations: Metz surveys 3 (ed. Chipot, M., Paulin, J. Saint Jean and Shafrir, I.). Pitman Research Notes in Mathematics, vol. 314, pp. 130–144 (Pitman, 1994).Google Scholar
8Gérard, P.. Microlocal defect measures. Commun. Partial Diffl Eqns 16 (1991), 1761–1794.CrossRefGoogle Scholar
9Glassey, R. T.. The cauchy problem in kinetic theory (SIAM, 1996).CrossRefGoogle Scholar
10Joly, J. L., Métivier, G. and Rauch, J.. Resonant one-dimensional nonlinear geometric optics. J. Functional Analysis 114 (1993), 106–231.CrossRefGoogle Scholar
11Joly, J. L., Métivier, G. and Rauch, J.. Trilinear compensated compactness and nonlinear geometric optics. Ann. Math. 142 (1995), 121–169.CrossRefGoogle Scholar
12Murat, F.. Compacité par compensation. Ann. Sci. Norm. Sup. Pisa 5 (1978), 489–507.Google Scholar
13McLaughlin, D., Papanicolaou, G. and Tartar, L.. Weak limits of semilinear hyperbolic systems with oscillating data. In Macroscopic modelling of turbulent flows (ed. Frisch, U., Keller, J. B., Papanicolaou, G. and Pironneau, O.). Springer Lecture Notes in Physics no. 230, pp. 277–289 (Springer, 1985).CrossRefGoogle Scholar
14Otto, F.. Dynamics of labyrinthine pattern formation in magnetic fluids: a mean-field theory. Arch. Ration. Mech. Analysis 141 (1998), 63–103.CrossRefGoogle Scholar
15Rachev, S. T.. Probability metrics and the stability of stochastic models (Wiley, 1991).Google Scholar
16Siegel, C. L. and Moser, J. K.. Lectures on celestial mechanics. Grundlehren, vol. 187 (Springer, 1971).CrossRefGoogle Scholar
17Tartar, L.. Compensated compactness and applications to PDEs. In Nonlinear mechanics and analysis (ed. Knops, R. J.). Pitman Reasearch Notes in Mathematics, vol. 4, pp. 136–212 (Pitman, 1979).Google Scholar
18Tartar, L.. Solutions oscillantes des equations de Carleman. Seminaire Goulaouic–Meyer–Schwartz, 1980–1981, Ecole Polytechnique (Palaiseau). Expsoe XII, 1981.Google Scholar
19Tartar, L.. The compensated compactness method applied to systems of conservation laws. In Systems of nonlinear PDE (ed. Ball, J. M.), pp. 263–285 (Reidel, D., 1983).Google Scholar
20Tartar, L.. Etude des oscillations dans thes equations aux derivees partielles non lineaires. In Trends in applications of pure mathematics to mechanics (ed. Roseau, M. and Ciarlet, P. G.). Springer Lecture Notes in Physics, no. 195, pp. 384–412 (Springer, 1984).CrossRefGoogle Scholar
21Tartar, L.. Oscillations and asymptotic behaviour of two semilinear hyperbolic systems. In Dynamics of infinite dimensional systems (ed. Chow, S. N. and Hale, J. K.), pp. 341–356 (Springer, 1987).CrossRefGoogle Scholar
22Tartar, L.. H-measures, a new approach for studying homogenisation, oscillations and concentrations effects in partial differential equations. Proc. R. Soc. Edinb. A 115 (1990), 193–230.CrossRefGoogle Scholar
23Tartar, L.. Oscillations for semilinear systems. Unpublished lecture notes, no. 91.3, 1991, 14 pp.Google Scholar
24Tartar, L.. Personal communication at Oberwolfach 1996.Google Scholar
25Theil, F.. Young-measure solutions for a microkinetically regularized wave equation. Draft. IfAM, Universität Hannover. September 1996.Google Scholar
26Theil, F.. Young measure solutions for a viscoelastically damped wave equation with nonmonotone stress–strain relation. Arch. Ration. Mech. Analysis 144 (1998), 47–78.CrossRefGoogle Scholar
27Theil, F. and Levitas, V. I.. A Hamiltonian model for martensitic phase transformations including microkinetic energy. Math. Mech. Solids. (Submitted.)Google Scholar