H-measures applied to symmetric systems

Abstract: "H-measures were recently introduced by Tartar [Thmo] as a tool that might provide much better understanding of propagating oscillations. Partial differential equations of mathematical physics can (almost always) be written in the form of a symmetric system: [n over [sigma] over k=1]A[superscript k][delta subscript k]u + Bu = f, where A[superscript k] and B are matrix functions, while u is a vector unknown function, and f a known vector function. In this work we prove a general propagation theorem for H-measures associated to symmetric systems (theorem 3). This result, combined with the localisation property ([Thmo]) is then used to obtain more precise results on the behaviour of H-measures associated to the wave equation and Maxwell's system. Particular attention is paid to the equations that change type: Tricomi's equation and variants. The H-measure is not supported in the elliptic region; it moves along the characteristics in the hyperbolic region, and bounces of [sic] the parabolic boundary, which separates the hyperbolic region from the elliptic region."