Abstract
We consider the compressible isentropic Euler equations on with a pressure law , where . 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 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 near a vacuum; thirdly, assuming to be quasinearly subharmonic near a vacuum; and finally, by assuming that and are Hölder continuous. We then extend these results to show global energy conservation for the domain where is bounded with a boundary. We show that we can extend these results to the compressible Navier–Stokes equations, even with degenerate viscosity.
Citation
Ibrokhimbek Akramov. Tomasz Dębiec. Jack Skipper. Emil Wiedemann. "Energy conservation for the compressible Euler and Navier–Stokes equations with vacuum." Anal. PDE 13 (3) 789 - 811, 2020. https://doi.org/10.2140/apde.2020.13.789
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