The 3D elastic wave system for admissible harmonic materials is shown to be locally well-posed with the divergence part in H^{3+} and the curl part in H^{4+}, the first low-regularity result for multi-wave-speed systems.
Parametrix for wave equations on a rough background II: construction and control at initial time
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abstract
This is the second of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} \phi=0$, where ${\bf g}$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000}, and solved by S. Klainerman, I. Rodnianski and the author in \cite{boundedl2}. On a more general level, this sequence of papers deals with the control of the eikonal equation on a rough background, and with the derivation of $L^2$ bounds for Fourier integral operators on manifolds with rough phases and symbols, and as such is also of independent interest.
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2024 1verdicts
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Low-Regularity Local Well-Posedness for the Elastic Wave System
The 3D elastic wave system for admissible harmonic materials is shown to be locally well-posed with the divergence part in H^{3+} and the curl part in H^{4+}, the first low-regularity result for multi-wave-speed systems.