Energy conservation for the compressible Euler and Navier–Stokes equations with vacuum
Energy conservation for the compressible Euler and Navier–Stokes equations with vacuum
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DOI：
10.2140/apde.2020.13.789
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发表时间：
2018-08
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影响因子：
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通讯作者：
Ibrokhimbek Akramov;Tomasz Dkebiec;Jack W. D. Skipper;E. Wiedemann
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作者：
Ibrokhimbek Akramov;Tomasz Dkebiec;Jack W. D. Skipper;E. Wiedemann
我们考虑在[0，T]$上的可压缩等熵欧拉方程，在C^{1，γ-1}中具有压力律$p，其中$1，Gamma<；2$.这包括所有物理上相关的情况，例如单原子气体。我们研究了在什么条件下弱解的正则性守恒。以前的结果已经很关键地假设了在密度范围内，C^2中的$p，然而，对于现实的压强定律，这意味着我们必须排除真空的情况。在这里，我们改进了这些结果，给出了能量守恒的一些充分条件，即使对于可能呈现真空的解：首先，假设速度是散度场；其次，在真空附近对$1/Rho$施加额外的可积性；第三，假设$u$在真空附近是拟近次调和的；最后，假设$u$和$\rho$是Holder连续的。然后我们将这些结果推广到区域$\Omega\Times[0，T]$中的全局能量守恒，其中$\Omega$有界于$C^2$边界。我们证明了我们可以将这些结果推广到可压缩的Navier-Stokes方程，即使具有简并的粘性。
We consider the compressible isentropic Euler equations on $\mathbb{T}^d\times [0,T]$ 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 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 $\Omega\times [0,T]$ 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.