#### Vol. 13, No. 3, 2020

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

### Ibrokhimbek Akramov, Tomasz Dębiec, Jack Skipper and Emil Wiedemann

Vol. 13 (2020), No. 3, 789–811
##### Abstract

We consider the compressible isentropic Euler equations on $\left[0,T\right]×{\mathbb{𝕋}}^{d}$ 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 $\left[0,T\right]×\Omega$ where $\Omega$ 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.

##### Keywords
compressible Euler equations, compressible Navier–Stokes equations, vacuum, Onsager's conjecture, energy conservation
##### Mathematical Subject Classification 2010
Primary: 35Q31
Secondary: 35Q30, 35L65, 76N10