Divergence-free symmetric tensors in magneto-hydrodynamics

In a bounded time-space domain of boldsymbol{M} space dimensions (boldsymbol{M}=2 or boldsymbol{M}=3) endowed with EUCLIDean metric, conservation of mass, conservation of momentum and the induction equation for the magnetic field are written for an ideal (inviscid) fluid, which is quasi-neutral, has constant electric conductivity and is subject to a conservative force of position-dependent potential varphi. Out of the system of 1+2boldsymbol{M} partial differential equations, those for {mass, momentum} formally correspond to the {time, space} divergence (Div[·]) of an Ntimes N tensor A which is real valued and symmetric. By assuming the existence of a solution {rho {f}, p,vec{u},vec{H}} ≡ {fluid density, pressure, velocity, magnetic field} to the system, conditions are provided for positive definiteness of A. From recent results by D. SERRE ['Divergence-free positive symmetric tensors and fluid dynamics,' Ann. I. H. Poincaré - Analyse Non-lin., vol. AN 35, pp. 1209-1234, 2018] estimates for the determinant, det [mathbf{A}], are derived in both cases, text{Div}[mathbf{A}]=vec{0} {N} and{equiv}vec{0} {N}. In the latter case, integrability gain consists of the property rho {f} {1/M}Phiin L {1}(Q), where Phi:=p+rho {f}varphi.