Higher integrability of a determinant related to a system of magneto-hydrodynamics

A system of partial differential equations models the magneto-hydrodynamics of an ideal fluid in a bounded time-space domain Q = (0, τ ) × BR [⃗0] 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 AEULER, which is real valued, symmetric and positive definite, provided the terms depending on the magnetic field, H⃗ , 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[AEULER] is investigated. Articles by D. SERRE from 2018 onwards [Divergence-free positive symmetric tensors and fluid dynamics. Ann. I. H. Poincare ́ - Analyse Non-lin.. 2018;AN 35:1209-1234] motivated this work. Letting Φ := p + ρf φ, with p pressure, ρf density and φ (>0) the potential of external forces, one has det[AEULER] = ΦMρf. If the entries of AEULER are in L1(Q), then higher integrability consists of ρ1/MΦ ∈ L1(Q) and holds provided Div[AEULER] has finite total mass in Q. Higher integrability thus affects p and ρf , leaving out velocity, ⃗u, and H⃗ . If a solution {ρf , p, ⃗u, H⃗ } were known to exist and comply with a number of requirements, then higher integrability would mean continuous dependence of ρ1/M Φ on H⃗ . In the absence of information about existence, the L1 (Q)-estimate is a qualitative result about the dynamical system: a property of the solution is inferred from that of det[AEULER]. Many problems are still open, some of which are stated.