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 \({\underline{B}}^{\alpha }_{p,VMO}\) ${\underline{B}}_{p,VMO}^{\alpha }$ and Besov spaces \(B^{\alpha }_{p,\infty }\) $B_{p,\infty }^{\alpha }$ 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.