Energy conservation for the compressible Euler and Navier–Stokes equations with vacuum

Akramov, Ibrokhimbek; Dębiec, Tomasz; Skipper, Jack; Wiedemann, Emil

doi:10.2140/apde.2020.13.789

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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

DOI: 10.2140/apde.2020.13.789

Abstract

We consider the compressible isentropic Euler equations on $[0, T] \times 𝕋^{d}$ with a pressure law $p \in C^{1, γ - 1}$ , where $1 \leq γ < 2$ . 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 $p \in C^{2}$ 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 $1 ∕ ρ$ near a vacuum; thirdly, assuming $ρ$ to be quasinearly subharmonic near a vacuum; and finally, by assuming that $u$ and $ρ$ are Hölder continuous. We then extend these results to show global energy conservation for the domain $[0, T] \times Ω$ where $Ω$ 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.

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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

Vol. 13, No. 3, 2020