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The global solvability of the Hall-magnetohydrodynamics system in critical Sobolev spaces

Raphaël Danchin

Université Paris-Est Créteil, LAMA UMR 8050, 61 Avenue du Général de Gaulle, 94010 Créteil, France

Sorbonne Université, LJLL UMR 7598, 4 Place Jussieu, 75005 Paris, France

E-mail Address: raphael.danchin@u-pec.fr

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and

Jin Tan

Université Paris-Est Créteil, LAMA UMR 8050, 61 Avenue du Général de Gaulle, 94010 Créteil, France

E-mail Address: jin.tan@u-pec.fr

Corresponding author.

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https://doi.org/10.1142/S0219199721500991Cited by:6 (Source: Crossref)

Abstract

We are concerned with the 3D incompressible Hall-magnetohydrodynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita–Kato’s theorem [On the Navier–Stokes initial value problem I, Arch. Ration. Mech. Anal. 16 (1964) 269–315] for the Navier–Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita–Kato regularity class. A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2 $\frac{1}{2}$ $\frac{1}{2}$ D flows for the Hall-MHD system (that is, 3D flows independent of the vertical variable), and establish the global existence of strong solutions, assuming only that the initial magnetic field is small. Our proofs strongly rely on the use of an extended formulation involving the so-called velocity of electron $v = u - ε \nabla \times B$ $v = u - 𝜀 \nabla \times B$ and as regards $2 \frac{1}{2}$ $2 \frac{1}{2}$ D flows, of the auxiliary vector-field $Ω = ε ω + B$ $Ω = 𝜀 ω + B$ that comes into play in the total magneto-helicity balance for the Hall-MHD system.

Keywords:

AMSC: 35Q35, 76D03, 86A10

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