Singular pseudodifferential calculus for wavetrains and pulses

TL;DR

This paper develops a novel singular pseudodifferential calculus that handles nonstandard decay in frequency variables, enabling analysis of highly oscillatory solutions in nonlinear hyperbolic problems and geometric optics.

Contribution

It introduces a new symbolic calculus for symbols lacking standard decay, with a regularization effect in anisotropic Sobolev spaces, extending previous results in nonlinear hyperbolic PDEs.

Findings

Established a regularization effect for remainders in the calculus.

Extended existence results for highly oscillatory solutions.

Applied to nonlinear geometric optics with boundary amplification.

Abstract

We develop a singular pseudodifferential calculus. The symbols that we consider do not satisfy the standard decay with respect to the frequency variables. We thus adopt a strategy based on the Calderon-Vaillancourt Theorem. The remainders in the symbolic calculus are bounded operators on $L^2$, whose norm is measured with respect to some small parameter. Our main improvement with respect to an earlier work by Williams consists in showing a regularization effect for the remainders. Due to a nonstandard decay in the frequency variables, the regularization takes place in a scale of anisotropic, and singular, Sobolev spaces. Our analysis allows to extend previous results on the existence of highly oscillatory solutions to nonlinear hyperbolic problems. The results are also used in a companion work to justify nonlinear geometric optics with boundary amplification, which corresponds to a more singular regime than any other one considered before. The analysis is carried out with either an additional real or periodic variable in order to cover problems for pulses or wavetrains in geometric optics.