Propagation of singularities for rough metrics

Abstract

We use a wave packet transform and weighted norm estimates in phase space to establish propagation of singularities for solutions to time-dependent scalar hyperbolic equations that have coefficients of limited regularity. It is assumed that the second order derivatives of the principal coefficients belong to L 1 t L ∞ x , and that u is a solution to the homogeneous equation of global Sobolev regularity s 0 = 0 or 1. It is then proven that the H s 0 +1 wavefront set of u is a union of maximally extended null bicharacteristic curves.

$L^{p}$ and $\mathcal{H}^{p}_{FIO}$ regularity for wave equations with rough coefficients, Part I

“…In fact, the Strichartz estimates obtained by Smith and Tataru using similar methods also hold for time-dependent coefficients, and one may even allow for time dependence of slightly lower regularity than the spatial regularity (see also [23]). The same applies to propagation of singularities [36], and our results can also be extended to coefficients that depend on time in a rough sense. However, doing so introduces various subtle technical issues that tend to obfuscate the main ideas for anyone who is not already familiar with them.…”

“…Accordingly, the flow parametrix does not arise from an actual bicharacteristic flow on phase space. We believe that it is useful to construct an anisotropic flow parametrix on phase space involving genuine bicharacteristic flows, both for technical reasons and for conceptual simplicity, as the author himself did using isotropic transforms in his later work [35,36].…”

“…Combining paradifferential methods due to Bony [5] with wave packet transforms, he constructed a useful microlocal parametrix on L 2 (R n ) and then corrected to the exact solution using an iterative procedure. This strategy was exploited by Smith, as well as Tataru, to obtain powerful results on rough wave equations, such as Strichartz estimates [34,[41][42][43], propagation of singularities [36], and the related spectral cluster estimates [35].…”

Off-singularity bounds and Hardy spaces for Fourier integral operators

“…They have also been used in the study of wave equations with rough coefficients by e.g. Smith [36,37] and Tataru [39,40]. Our notion of off-singularity decay appears in the work of Smith [34,35], and similar concepts have been used in the numerical analysis of wave equations [10] and in the time-frequency analysis of Schrödinger operators [15].…”

Rough pseudodifferential operators on Hardy spaces for Fourier integral operators

“…The resulting parametrix then contains a lot of information about the rough equation, while also being more tractable than the bona fide solution operators to the equation. A specific instance of this paradigm was developed by Smith in [20] and subsequently applied by both Smith and Tataru to obtain powerful results for wave equations with rough coefficients, such as Strichartz estimates [20,[23][24][25], propagation of singularities [22], and the related spectral cluster estimates [21].…”