# Energy conservation for the weak solutions to the equations of compressible magnetohydrodynamic flows in three dimensions

## TL;DR

This paper proves energy conservation for weak solutions of 3D compressible MHD flows under specific regularity conditions on density and velocity, extending previous results to include magnetic fields.

## Contribution

It extends energy conservation results from compressible Navier-Stokes to magnetohydrodynamic flows, requiring only regularity conditions on density and velocity.

## Key findings

- Energy conservation holds under certain regularity conditions.
- Magnetic field inclusion does not alter the regularity requirements.
- Results generalize previous work on Navier-Stokes equations.

## Abstract

In this paper, we prove the energy conservation for the weak solutions to the three-dimensional equations of compressible magnetohydrodynamic flows (MHD) under certain conditions only about density and velocity. This work is inspired by the seminal work by Yu [27] on the energy conservation of compressible Navier-Stokes equations. Our result indicates that even the magnetic field is taken into account, we only need some regularity conditions of the density and velocity as in [27] to ensure the energy conservation.