Energy conservation for the weak solutions to the 3D compressible magnetohydrodynamic equations of viscous non-resistive fluids in a bounded domain

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Abstract

In this paper, inspired by the work of Chen–Liang–Wang–Xu (Chen et al., 2020) on compressible Navier–Stokes equations, we obtain the energy conservation for weak solutions of the compressible non-resistive magnetohydrodynamic flows in a bounded domain Ω⊂R3. To ensure the energy conservation, we need the same regularity conditions of density and velocity as in Chen et al. (2020), moreover, H∈Lt4Lx4 and ∇H∈Ltp1Lxq1, where p1=4p3p−4, q1=4q3q−4. Under these conditions, p1∈(43,2], q1∈(43,127] when p≥4 and q≥6.

Section snippets

Preliminaries

The following notations and results will be used in the proof of Theorem 1.1.

Proof of Theorem 1.1

Throughout the rest of the paper, we denote ∫Ωf≔∫Ωfdx, ∫0T∫Ωf≔∫0T∫Ωfdxdt. First of all, we can rewrite model (1.2) as follows ρt+div(ρu)=0,∂t(ρu)+div(ρu⊗u)+∇ργ=div(H⊗H)−∇(|H|22)+μΔu+(λ+μ)∇divu,Ht−div(u⊗H)+div(H⊗u)=0,divH=0.Then, we use the method which is mentioned in Section 2.

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    Approximation on ∂Ω. We select ηε(xε−y,t−s) as the test function. Taking the jth component of the first term div(H⊗H) on the right-hand side of (3.1)2 for example, we have ∫0T∫Vi−Lεn→(xi)divy(Hj⊗H)(y,s)ηε(xε−y,t−s)dyds=∫0

Acknowledgments

The authors would like to thank the anonymous referees for providing insightful comments and constructive suggestions. They also thank Professor Huanyao Wen and Mr. Yinghui Wang for valuable discussions.

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