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.

Crosta, G. (2021). Higher integrability of a determinant related to a system of magneto-hydrodynamic equations [Working paper].

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

Crosta, GFF
2021

Abstract

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.
Working paper
Capitolo accettato per la pubblicazione.
positive definite symmetric tensors, tensor divergence, determinant, integrability gain, magnetohydrodynamics, Euclidean metric, ideal fluid, quasi-neutral fluid, induction equation, conservation laws
English
2021
1
18
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Crosta, G. (2021). Higher integrability of a determinant related to a system of magneto-hydrodynamic equations [Working paper].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/360678
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