Validity of nonlinear geometric optics for entropy solutions of multidimensional scalar conservation laws

Gui-Qiang Chen (1) , Stéphane Junca (2) , Michel Rascle (2)

1 Northwestern University [Evanston]
2 LJAD - Laboratoire Jean Alexandre Dieudonné

Nonlinear geometric optics with various frequencies for entropy solutions only in $L^∞$ of multidimensional scalar conservation laws is analyzed. A new approach to validate nonlinear geometric optics is developed via entropy dissipation through scaling, compactness, homogenization, and $L^1$-stability. New multidimensional features are recognized, especially including nonlinear propagations of oscillations with high frequencies. The validity of nonlinear geometric optics for entropy solutions in $L^∞$ of multidimensional scalar conservation laws is justified.

Mots clés: BV; Perturbation; Validity; Nonlinear geometric optics; Multidimensional conservation laws; Profile; Entropy solutions; Entropy dissipation; Compactness; Homogenization; Oscillation; Scaling; Stability; Multiscale

Domaines: Equations aux dérivées partielles [math.AP]

Gui-Qiang Chen, Stéphane Junca, Michel Rascle. Validity of nonlinear geometric optics for entropy solutions of multidimensional scalar conservation laws. Journal of Differential Equations, 2006, 222 (2), pp.439-475. ⟨10.1016/j.jde.2005.04.014⟩. ⟨hal-01312334⟩