Singular Pseudodifferential Operators, Symmetrizers, and Oscillatory Multidimensional Shocks

Abstract

We introduce a calculus of singular pseudodifferential operators (SPOs) depending on wavelength ε and use them to solve three different types of singular quasilinear hyperbolic systems. Such systems arise in nonlinear geometric optics and also, for example, in the study of incompressible limits and of nonlinear wave equations with small nonlinear terms or small data. The SPOs act in both slow and fast variables and are singular not only because their symbols have finite regularity and depend on 1ε, but also because their derivatives fail to decay in the usual way in the dual variables. There is a necessarily crude calculus with large parameter (e.g., residual operators are just bounded on L2), but the calculus admits the proof of Garding inequalities and enables us to symmetrize and sometimes even diagonalize the singular systems being considered by microlocalizing simultaneously in both slow and fast variables. The paper culminates in a proof of the existence of oscillatory multidimensional shocks on a fixed time interval independent of the wavelength ε as ε→0. The use of SPOs allows us to eliminate the small divisor assumptions made in earlier work and also to construct more general oscillatory solutions in which elliptic boundary layers are present on one or both sides of the shock.