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 time-space domain Q =(0, \tau) × B_R[O] endowed with Euclidean metric and having M=3 space dimensions. The equations describe conservation of mass, conservation of momentum and magnetic induction. The {mass, momentum} equations correspond to the {time, space} divergence (Div[·]) of a 4 × 4 tensor denoted A_{Euler}, which is real valued, symmetric and positive definite, provided the terms depending on the magnetic field, \vec 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 det[A_{Euler}] is investigated. Articles by D. SERRE from 2018 onwards [Divergence-free positive symmetric tensors and fluid dynamics. Ann. I. H. Poincaré - Analyse Non-lineaire. 2018; AN 35:1209-1234] motivated this work. Letting Φ := p + ρ_f φ, with p pressure, ρ_f density and φ (>0) the potential of external forces, one has det[A_{Euler}] = Φ^M ρ_f . If the entries of A_{Euler} are in L^1(Q), then higher integrability consists of ρ_f^{1/M}Φ ∈ L^1(Q) and holds provided Div[A_{Euler}] has finite total mass in Q. Higher integrability thus affects p and ρ_f , leaving out velocity, \vec u and \vec H. If a solution {ρ_f , p, \vec u, \vec H } were known to exist and comply with a number of requirements, then higher integrability would mean continuous dependence of ρ_f^{1/M}Φ on \vec H. In the absence of information about existence, the L^1(Q)-estimate is a qualitative result about the dynamical system: a property of the solution is inferred from that of det[A_{Euler}]. Many problems are still open, some of which are stated.