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

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.