Energy conservation for the compressible euler and navier-stokes equations with vacuum

Akramov I, Debiec T, Skipper J, Wiedemann E (2020)

Publication Type: Journal article

Publication year: 2020

Journal

Analysis & Pde

Book Volume: 13

Pages Range: 789-811

Journal Issue: 3

DOI: 10.2140/apde.2020.13.789

Abstract

We consider the compressible isentropic Euler equations on [0, T]xT^d with a pressure law p ∈ C^{1, 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.

Authors with CRIS profile

Emil Wiedemann

Involved external institutions

Universität Ulm

Germany (DE) University of Warsaw / Uniwersytet Warszawski

Poland (PL)

How to cite

APA:

Akramov, I., Debiec, T., Skipper, J., & Wiedemann, E. (2020). Energy conservation for the compressible euler and navier-stokes equations with vacuum. Analysis & Pde, 13(3), 789-811. https://dx.doi.org/10.2140/apde.2020.13.789

MLA:

Akramov, Ibrokhimbek, et al. "Energy conservation for the compressible euler and navier-stokes equations with vacuum." Analysis & Pde 13.3 (2020): 789-811.

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