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• W2886165619 endingPage "811" @default.
• W2886165619 startingPage "789" @default.
• W2886165619 abstract "We consider the compressible isentropic Euler equations on $mathbb{T}^dtimes [0,T]$ with a pressure law $pin C^{1,gamma-1}$, where $1le gamma <2$. 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 $pin C^2$ 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 quasi-nearly subharmonic near a vacuum; and finally, by assuming that $u$ and $rho$ are Holder continuous. We then extend these results to show global energy conservation for the domain $Omegatimes [0,T]$ 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." @default.
• W2886165619 created "2018-08-22" @default.
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• W2886165619 date "2020-04-15" @default.
• W2886165619 modified "2023-10-16" @default.
• W2886165619 title "Energy conservation for the compressible Euler and Navier–Stokes equations with vacuum" @default.
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• W2886165619 doi "https://doi.org/10.2140/apde.2020.13.789" @default.
• W2886165619 hasPublicationYear "2020" @default.
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