Vol. 13, No. 3, 2020

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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

Vol. 13 (2020), No. 3, 789–811
Abstract

We consider the compressible isentropic Euler equations on $\left[0,T\right]×{\mathbb{𝕋}}^{d}$ with a pressure law $p\in {C}^{1,\gamma -1}$, where $1\le \gamma <2$. 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 $\left[0,T\right]×\Omega$ 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.

Keywords
compressible Euler equations, compressible Navier–Stokes equations, vacuum, Onsager's conjecture, energy conservation
Mathematical Subject Classification 2010
Primary: 35Q31
Secondary: 35Q30, 35L65, 76N10
Milestones
Received: 16 August 2018
Revised: 18 December 2018
Accepted: 25 March 2019
Published: 15 April 2020
Authors
 Ibrokhimbek Akramov Institute of Applied Analysis Ulm University Ulm Germany Tomasz Dębiec Institute of Applied Mathematics and Mechanics University of Warsaw Warsaw Poland Jack Skipper Institute of Applied Analysis Ulm University Ulm Germany Emil Wiedemann Institute of Applied Analysis Ulm University Ulm Germany