Energy conservation for the compressible Eulerand Navier–Stokes equations with vacuum
Ibrokhimbek Akramov, Tomasz Dębiec, Jack W. D. Skipper, Emil Wiedemann
Abstract
We consider the compressible isentropic Euler equations on [math] with a pressure law [math] , where [math] . 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 [math] 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 [math] near a vacuum; thirdly, assuming [math] to be quasinearly subharmonic near a vacuum; and finally, by assuming that [math] and [math] are Hölder continuous. We then extend these results to show global energy conservation for the domain [math] where [math] is bounded with a [math] boundary. We show that we can extend these results to the compressible Navier–Stokes equations, even with degenerate viscosity.