We study the elastic wave system in three spatial dimensions. For admissible harmonic elastic materials, we prove a desired low-regularity local well-posedness result for the corresponding elastic wave equations. For such materials, we can split the dynamics into the “divergence-part” and the “curl-part,” and each part satisfies a distinct coupled quasilinear wave system with respect to different acoustical metrics. Our main result is that the Sobolev norm \(H^{3+}\) of the “divergence-part” (the “faster-wave part”) and the \({H^{4 + }}\) of the “curl-part” (the “slower-wave part”) can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption \(H^{3+}\) is optimal for the “divergence-part.” This marks the first favorable low-regularity local well-posedness result for a wave system with multiple wave speeds. Compared to the quasilinear wave equation, new difficulties arise from the multiple wave-speed nature of the system. Specifically, the acoustic metric \(\mathbf{g}\) of the faster-wave depends on both the faster-wave and slower-wave parts. Additionally, the dynamics of the faster-wave “divergence-part” require higher regularity of the “curl-part”. In particular, the Ricci curvature associated with the faster-wave is one derivative rougher than that of the slower-wave dynamics.This phenomenon also appears in the compressible Euler equations (featuring multiple characteristic speeds) and is a major obstacle to obtaining low-regularity local well-posedness results for general quasilinear wave systems if the two parts do not exhibit strong decoupling properties or if the “curl-part” lacks the structure necessary for better regularity results. For the elastic wave system governing the dynamics of the admissible harmonic elastic materials, we report that we can overcome these difficulties. For this system, we exploit its geometric structures and find that the “divergence-part” and “curl-part” exhibit decoupling properties and both parts show regularity gains. Moreover, we prove that the “divergence-part” maintains to represent the faster-wave throughout the entire time of the existence of the solution, ensuring that the characteristic hypersurfaces of the faster-wave are spacelike with respect to the slower-wave. This implies a crucial coerciveness for the geometric cone-flux energy of the “curl-part” on such characteristic hypersurfaces of the “divergence-part.F We furthermore carefully address all these challenges through spacetime energy estimates, Strichartz estimates, frequency-localized decay estimates, and conformal energy estimates. In all these estimates, we also precisely trace the impact of the “curl-part” on the faster-wave dynamics and control the associated geometry via employing the vector field method and the Littlewood-Paley theory.
