Global well-posedness and vanishing viscosity limit of the compressible elastic system in three dimensions

The compressible elastodynamics is a typical example of systems with different wave speeds, which are difficult to be solved due to lack of symmetries. In general, the nonlinear interactions among the pressure waves are so strong that the global existence of classical solutions cannot be expected. In this article we investigate a specific example, namely the compressible Mooney–Rivlin materials, of which both the interactions among the pressure waves and among the shear waves satisfy the null conditions respectively. With delicate analysis of the linear system, we manage to identify all the good unknowns in a simpler form which are essential to exploit the null conditions for extra time decay. The approach to the a priori energy estimates applies to both inviscid and viscous systems, and enables us to justify the vanishing viscosity limit result.