Energy conservation for the compressible euler and navier-stokes equations with vacuum - Akramov, Ibrokhimbek; Debiec, Tomasz; Skipper, Jack; Wiedemann, Emil (Analysis & Pde, 2020)
We consider the compressible isentropic Euler equations on [0, T]xTd with a pressure law p ∈ C1, γ-1, where 1 ≤ γ < 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 ∈ C2 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/ρ near a vacuum; thirdly, assuming ρ to be quasinearly subharmonic near a vacuum; and finally, by assuming that u and ρ are Holder continuous. We then extend these results to show global energy conservation for the domain [0, T]xΩ where Ω is bounded with a C2 boundary. We show that we can extend these results to the compressible Navier-Stokes equations, even with degenerate viscosity.
CRIS-Team Scopus Importer:2023-01-17
Keywords: Compressible euler equations; Compressible navier-stokes equations; Energy conservation; Onsager's conjecture; Vacuum