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Energy conservation
for the compressible Euler and Navier–Stokes equations with
vacuum
Ibrokhimbek Akramov, Tomasz Dębiec, Jack Skipper and Emil Wiedemann |
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Vol. 13 (2020), No. 3, 789–811
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Abstract |
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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. |
Keywords
compressible Euler equations, compressible Navier–Stokes
equations, vacuum, Onsager's conjecture, energy
conservation
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Mathematical Subject Classification 2010
Primary: 35Q31
Secondary: 35Q30, 35L65, 76N10
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Milestones
Received: 16 August 2018
Revised: 18 December 2018
Accepted: 25 March 2019
Published: 15 April 2020
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Authors |
| Ibrokhimbek Akramov | |
| Institute of Applied Analysis Ulm University Ulm Germany |
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| Tomasz Dębiec | |
| Institute of Applied Mathematics and
Mechanics University of Warsaw Warsaw Poland |
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| Jack Skipper | |
| Institute of Applied Analysis Ulm University Ulm Germany |
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| Emil Wiedemann | |
| Institute of Applied Analysis Ulm University Ulm Germany |
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