Energy Conservation for the Compressible Euler and Navier-Stokes Equations with Vacuum

We consider the compressible isentropic Euler equations on $\mathbb{T}^d\times [0,T]$ with a pressure law $p\in C^{1,γ-1}$, where $1\le γ<2$. 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 $1/ρ$ near a vacuum; thirdly, assuming $ρ$ to be quasi-nearly subharmonic near a vacuum; and finally, by assuming that $u$ and $ρ$ are Hölder continuous. We then extend these results to show global energy conservation for the domain $Ω\times [0,T]$ where $Ω$ 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.

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