Supercritical Geometric Optics for multidimensional scalar conservation laws

This article studies the propagation, near a smooth non-constant ground state, of monophase oscillatory wave trains with small amplitude and very high frequency for multidimensional nonlinear scalar conservation laws. These oscillations lie beyond the classical regime of uniformly bounded variation. Two scenarios may occur. Under suitable compatibility conditions, the oscillations propagate. Otherwise, they are damped through shock formation. In the baseline supercritical regime, we give an explicit criterion distinguishing these two behaviors and justify the corresponding asymptotic descriptions. At higher frequencies, the propagation mechanism becomes more restrictive and is governed by compatibility conditions involving a Lie algebra. Under a natural local constant-rank assumption, we obtain a necessary and sufficient condition for the existence of complete WKB expansions. We also prove cancellation under first-order incompatibility. Perturbing a non-constant background state limits the propagation of small scales compared with the constant-background case. Our constructions contribute to the understanding of the fractional smoothing effect conjectured by Lions, Perthame and Tadmor.