Higher integrability of a determinant related to a system of magneto-hydrodynamic equations

A system of partial differential equations models the magneto-hydrodynamics of an ideal fluid in a bounded domain $Q= (0, au) imes B_R[ec 0]$ endowed with extsc{Euclid}ean metric and having extit{M}=3 space dimensions. The equations describe conservation of mass and momentum, and magnetic induction. The {mass, momentum} equations correspond to the {time, space} divergence (${ m Div}[cdot]$) of a $4 imes 4$ tensor denoted ${f A}_{ m{ extsc{Euler}}}$, which is real valued, symmetric and positive definite, provided the terms depending on the magnetic field, $ec H$, are moved to the right side. By assuming the existence of a solution to some initial-boundary value problem, a property known as the ``higher integrability'' of ${ m det}[{f A}_{ m{ extsc{Euler}}}]$ is investigated. Articles by extsc{D. Serre} from 2018 onwards [Divergence-free positive symmetric tensors and fluid dynamics. extit{Ann. I. H. Poincar'e - Analyse Non-lin.}. 2018; extbf{AN 35}:1209-1234] motivated this work. Letting $Phi:= p + ho_f arphi$, with $p$ pressure, $ ho_f$ density and $arphi (>$0) the potential of external forces, one has ${ m det}[{f A}_{ m{ extsc{Euler}}}] = Phi^M ho_f$. If the entries of ${f A}_{ m{ extsc{Euler}}}$ are in $L^1(Q)$, then higher integrability consists of $ ho_f ^{1/M}Phi in L^1(Q)$ and holds provided ${ m Div}[{f A}_{ m{ extsc{Euler}}}]$ has finite total mass in $Q$. Higher integrability affects $p$ and $ ho_f$. If a solution ${ ho_f, p, ec u, ec H}$ were known to exist, then higher integrability would mean continuous dependence of $ ho_f ^{1/M}Phi$ on $ec H$. In the absence of information about existence, the $L^1(Q)$-estimate is a qualitative result about the dynamical system. Some open problems are stated.