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On the static and dynamic study of oscillations for some nonlinear hyperbolic systems of conservation laws. (English) Zbl 0768.35060

The main result for a \(2\times 2\) hyperbolic system having two linearly degenerate eigenvalues is, that oscillations cannot appear, but if they are present, they can propagate. The proof is partly formal, relying on compensated compactness ideas and the characteristics, and presents insight into the technical, resp., open problems.
The extension to the system of gas dynamics in Eulerian coordinates is linked to the interesting question of the ‘separation of the wave cone and the constitutive manifold’. This reflects ideas of Ron DiPerna.
As a conclusion: The property of the weak version of ‘ellipticity’ explains why genuine nonlinear systems do not admit oscillating solutions, in contrast to the linearly degenerate case.

MSC:

35L65 Hyperbolic conservation laws
35A30 Geometric theory, characteristics, transformations in context of PDEs
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References:

[1] J. M. Ball, A Version of the Fundamental Theorem for Young Measures, in {\it PDEs and Continuum Models of Phase Transitions}, M. Rascle, D. Serre and M. Slemrod Ed., Springer Lecture Notes in Physics, No. 344, pp. 207-216. · Zbl 0991.49500
[2] Chen Gui-Qiang, Propagation and Cancellation of Oscillations for Hyperbolic Systems of Conservation laws, Preprint, 1990. · Zbl 0727.35085
[3] Dafermos, C. M., Heriot watt symposium, (Knops, R. J., Res. Notes in Math., Nonlinear Analysis and Mechanics, Vol. 1, (1979), Pitman)
[4] DiPerna, R. J., Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal., Vol. 82, 8, 27-70, (1983) · Zbl 0519.35054
[5] DiPerna, R. J., Measure-valued solutions to conservation laws, Arch. Rat. Mech. Anal., Vol. 88, 223-270, (1985) · Zbl 0616.35055
[6] R. J. DiPerna, Compensated Compactness and General Systems of Conservation Laws. · Zbl 0555.35087
[7] P. Gerard, Compacité par compensation et régularité 2-microlocale. Preprint. · Zbl 0707.35032
[8] Murat, F., L’injection du cône positif de H− dans W−1, q est compacte pour tout q < 2, J. Math. Pures et Appl., T. 60, 309-322, (1981) · Zbl 0471.46020
[9] Rascle, M., Perturbations par viscosité de certains systèmes hyperboliques non linéaires, (1983), Thèse Lyon
[10] Rascle, M., On the convergence of the viscosity method for the system of nonlinear 1-D elasticity, A.M.S. Lectures in Applied Math., Vol. 23, 359-377, (1986)
[11] Serre, D., La compacité par compensation pour LES systèmes non linéaires de deux équations à une dimension d’espace, J. Maths. Pures Appl., T. 65, 423-468, (1987) · Zbl 0601.35070
[12] D. Serre, Oscillations non linéaires des systèmes hyperboliques. Méthodes et résultats qualitatifs, this issue. · Zbl 0810.35060
[13] Tartar, L., Compensated compactness and applications to partial differential equations, (Knops, R. J., Heriot Watt Symposium, (1979), Pitman), {\it Res. Notes in Math., Nonlinear Analysis and Mechanics} · Zbl 0437.35004
[14] L. Tartar, H-Measures, a New Approach for Studying Homogenization and Concentration Effects in Partial Differential Equations, Preprint. · Zbl 0774.35008
[15] Wagner, D., Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Diff. Eqs., Vol. 68, 118-136, (1987) · Zbl 0647.76049
[16] L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, Saunders, Philadelphia, 1969. · Zbl 0177.37801
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