Energy Equality in Compressible Fluids with Physical Boundaries

Abstract

We study the energy balance for the weak solutions of the three-dimensional compressible Navier--Stokes equations in a bounded domain. By constructing a global mollification combined with an independent boundary cut-off and then taking a double limit to prove the convergence of the resolved energy, we establish an $L^{p}$-$L^{q}$ regularity condition on the velocity field for the energy equality to hold, provided that the density is bounded and satisfies $\sqrt{\rho} \in L^\infty_t H^1_x$. As a result of our new approach, we can avoid assuming additional regularity of the velocity near the boundary in order to deal with the boundary production due to the diffusion terms.

Topics & Concepts