## Energy conservation for the compressible Euler and Navier-Stokes equations with vacuum.(English)Zbl 1437.35552

Summary: We consider the compressible isentropic Euler equations on $$[0,T]\times\mathbb{T}^d$$ with a pressure law $$p\in C^{1,\gamma-1}$$, where $$1\leq \gamma <2$$. 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 $$[0,T]\times\Omega$$ 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.

### MSC:

 35Q31 Euler equations 35Q30 Navier-Stokes equations 35L65 Hyperbolic conservation laws 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35D30 Weak solutions to PDEs 35B65 Smoothness and regularity of solutions to PDEs
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