Energy conservation is proved for weak solutions of compressible MHD in 3D using only the density and velocity regularity conditions from prior Navier-Stokes work.
Energy equality in compressible fluids with physical boundaries
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abstract
We study the energy balance for weak solutions of the three-dimensional compressible Navier--Stokes equations in a bounded domain. We establish an $L^p$-$L^q$ regularity conditions on the velocity field for the energy equality to hold, provided that the density is bounded and satisfies $\sqrt{\rho} \in L^\infty_t H^1_x$. The main idea is to construct a global mollification combined with an independent boundary cut-off, and then take a double limit to prove the convergence of the resolved energy.
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Energy conservation for the weak solutions to the equations of compressible magnetohydrodynamic flows in three dimensions
Energy conservation is proved for weak solutions of compressible MHD in 3D using only the density and velocity regularity conditions from prior Navier-Stokes work.