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

Akramov, Ibrokhimbek; Dębiec, Tomasz; Skipper, Jack; Wiedemann, Emil

Anal. PDE 13, No. 3, 789-811 (2020).

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

Keywords:

compressible Euler equations; compressible Navier-Stokes equations; vacuum; Onsager’s conjecture; energy conservation