This research proposes a comprehensive framework for establishing global existence of solutions to the compressible elastodynamics equations in arbitrary spatial dimensions. It combines a refined null condition framework, dispersive estimates, polyconvex energy methods, and measure-valued solution concepts to address challenges in shock formation, material frame indifference, and the transition from compressible to incompressible limits.
Key findings
Global existence for small initial data in dimensions d≥4 without irrotationality assumptions.
Almost global existence in d=3, identifying the critical dimension d=2 where additional structural assumptions are required.
Hybrid numerical schemes combining spectral methods with high-order time integration validated against theoretical predictions.
Limitations & open questions
Fundamental questions regarding global well-posedness in higher dimensions remain open.
Challenges in higher dimensions include quasilinear hyperbolicity, the null condition, decay rates, constitutive constraints, and measure formation.