标题： 可压缩欧拉和纳维-斯托克斯方程在真空中能量守恒 显示英文标题

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

摘要： 我们研究了在具有压力定律$p\in C^{1,\gamma-1}$的$\mathbb{T}^d\times [0,T]$上的可压缩等熵欧拉方程组，其中$1\le \gamma <2$。这涵盖了所有物理相关的情况，例如单原子气体。我们探讨了在何种正则性条件下弱解会保持能量守恒。先前的结果至关重要地假设了$p\in C^2$在密度范围内，然而，对于现实的压力定律来说，这意味着我们必须排除真空情况。 这里我们通过给出若干充分条件来改进这些结果，即使对于可能出现真空的解也能保证能量守恒：首先，假设速度场为散度测度场；其次，在真空附近对 $1/\rho$ 施加额外的可积性条件；第三，在真空附近假设 $\rho$ 为拟近似次调和函数；最后，假设 $u$ 和 $\rho$ 是 Hölder 连续的。 然后我们将这些结果推广到证明当 $\Omega\times [0,T]$ 域内 $\Omega$ 有界且具有 $C^2$ 边界时的整体能量守恒。 我们表明可以将这些结果扩展到带有退化粘性的可压缩 Navier-Stokes 方程。 显示英文摘要

摘要： 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 H\"older 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.