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 dā„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.