On compactness estimates for hyperbolic systems of conservation laws

Ancona, Fabio; Glass, Olivier; Nguyen, Khai T.

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 6, pp. 1229-1257.

Nous étudions la compacité dans $L_{\mathrm{loc}}^1$ du semi-groupe ${\left(S_t\right)}_{t>0}$ définissant les solutions faibles d'entropie de systèmes hyperboliques de lois de conservation généraux en dimension un d'espace. Nous établissons une estimée inférieure de l'ε-entropie de Kolmogorov de l'image par l'application $S_t$ d'ensembles bornés dans $L^1\cap L^{\infty }$, qui est du même ordre $1/ϵ$ que celles establies par les auteurs pour les lois de conservation scalaires. Nous obtenons aussi une estimée supérieure d'ordre $1/ϵ$ pour l'ε-entropie de Kolmogorov de tels ensembles dans le cas des systèmes de Temple avec des champs charactéristiques vraiment non linéaires, ce qui étend le même type d'estimées obtenues par De Lellis et Golse dans le cas des lois de conservation scalaires à flux convexe. Comme suggéré par Lax, ces estimées quantitatives pourraient donner une mesure de l'ordre de « résolution » de méthodes numériques mises en place pour ces équations.

We study the compactness in $L_{\mathrm{loc}}^1$ of the semigroup mapping ${\left(S_t\right)}_{t>0}$ defining entropy weak solutions of general hyperbolic systems of conservation laws in one space dimension. We establish a lower estimate for the Kolmogorov ε-entropy of the image through the mapping $S_t$ of bounded sets in $L^1\cap L^{\infty }$, which is of the same order $1/ϵ$ as the ones established by the authors for scalar conservation laws. We also provide an upper estimate of order $1/ϵ$ for the Kolmogorov ε-entropy of such sets in the case of Temple systems with genuinely nonlinear characteristic families, that extends the same type of estimate derived by De Lellis and Golse for scalar conservation laws with convex flux. As suggested by Lax, these quantitative compactness estimates could provide a measure of the order of “resolution” of the numerical methods implemented for these equations.

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