Energy equality in compressible fluids with physical boundaries

Abstract

We study the energy balance for weak solutions of the three-dimensional compressible Navier--Stokes equations in a bounded domain. We establish an $L^p$-$L^q$ regularity conditions on the velocity field for the energy equality to hold, provided that the density is bounded and satisfies $\sqrt{\rho }\in L_t^{\infty }{H}_x^1$. The main idea is to construct a global mollification combined with an independent boundary cut-off, and then take a double limit to prove the convergence of the resolved energy.