Energy Conservation for the Compressible Euler Equations and Elastodynamics

In this paper, we consider the Onsager's conjecture for the compressible Euler equations and elastodynamics in a torus or a bounded domain. Some energy conservation criteria in Onsager's critical spaces (B) under bar (alpha)(p,VMO) and Besov spaces B-p,infinity(alpha) for weak solutions in these systems are established, which extend the known corresponding results. A novel ingredient is the utilization of a test function in one single step rather than two steps in the case of incompressible models to capture the affect of the boundary.