Energy conservation for the compressible Euler and Navier–Stokes equations with vacuum

Abstract

We consider the compressible isentropic Euler equations on $\left[0,T\right]\times {\mathbb{𝕋}}^d$ with a pressure law $p\in C^{1,\gamma -1}$, where $1\le \gamma <2$. This includes all physically relevant cases, e.g., the monoatomic gas. We investigate under what conditions on its regularity a weak solution conserves the energy. Previous results have crucially assumed that $p\in C^2$ in the range of the density; however, for realistic pressure laws this means that we must exclude the vacuum case. Here we improve these results by giving a number of sufficient conditions for the conservation of energy, even for solutions that may exhibit vacuum: firstly, by assuming the velocity to be a divergence-measure field; secondly, imposing extra integrability on $1∕\rho$ near a vacuum; thirdly, assuming $\rho$ to be quasinearly subharmonic near a vacuum; and finally, by assuming that $u$ and $\rho$ are Hölder continuous. We then extend these results to show global energy conservation for the domain $\left[0,T\right]\times \mathrm{\Omega }$ where $\mathrm{\Omega }$ is bounded with a $C^2$ boundary. We show that we can extend these results to the compressible Navier–Stokes equations, even with degenerate viscosity.