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Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion
Analysis of PDEs · Submitted 2 Apr 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This paper breaks the barrier and shows that the fractional Hall-MHD equations are locally well-posed for any $\alpha>\frac12$.
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The abstract, sentence by sentence
The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from kinetic models, are useful in describing many physical phenomena in geophysics and astrophysics. This paper studies the local well-posedness of classical solutions to the Hall-MHD equations with the magnetic diffusion given by a fractional Laplacian operator, $(-\Delta)^\alpha$. Due to the presence of the Hall term in the Hall-MHD equations, standard energy estimates appear to indicate that we need $\alpha\ge 1$ in order to obtain the local well-posedness. This paper breaks the barrier and shows that the fractional Hall-MHD equations are locally well-posed for any $\alpha>\frac12$. The approach here fully exploits the smoothing effects of the dissipation and establishes the local bounds for the Sobolev norms through the Besov space techniques. The method presented here may be applicable to similar situations involving other partial differential equations.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from kinetic models, are useful in describing many physical phenomena in geophysics and astrophysics. | background 1.00 | 0.06 | 0.01 | asserted 0.99 |
| s1 | This paper studies the local well-posedness of classical solutions to the Hall-MHD equations with the magnetic diffusion given by a fractional Laplacian operator, $(-\Delta)^\alpha$. | contribution 0.94 | 0.88 | 0.05 | asserted 0.89 |
| s2 | Due to the presence of the Hall term in the Hall-MHD equations, standard energy estimates appear to indicate that we need $\alpha\ge 1$ in order to obtain the local well-posedness. | gap 0.57 | 0.23 | 0.03 | hedged possibility 0.99 |
| s3 | This paper breaks the barrier and shows that the fractional Hall-MHD equations are locally well-posed for any $\alpha>\frac12$. | result 0.91 | 0.97 | 0.95 | asserted 1.00 |
| s4 | The approach here fully exploits the smoothing effects of the dissipation and establishes the local bounds for the Sobolev norms through the Besov space techniques. | method 1.00 | 0.94 | 0.03 | asserted 1.00 |
| s5 | The method presented here may be applicable to similar situations involving other partial differential equations. | interpretation 0.56 | 0.20 | 0.02 | hedged possibility 1.00 |
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The paper
Original abstract
The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from kinetic models, are useful in describing many physical phenomena in geophysics and astrophysics. This paper studies the local well-posedness of classical solutions to the Hall-MHD equations with the magnetic diffusion given by a fractional Laplacian operator, $(-\Delta)^\alpha$. Due to the presence of the Hall term in the Hall-MHD equations, standard energy estimates appear to indicate that we need $\alpha\ge 1$ in order to obtain the local well-posedness. This paper breaks the barrier and shows that the fractional Hall-MHD equations are locally well-posed for any $\alpha>\frac12$. The approach here fully exploits the smoothing effects of the dissipation and establishes the local bounds for the Sobolev norms through the Besov space techniques. The method presented here may be applicable to similar situations involving other partial differential equations.
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How the literature uses it
10 arXiv papers cite this in 10 sentences
Papers citing it · 1 sentence without a confident label
Cites it as background 9 papers
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Recently, there are many researches on the Hall-MHD equations, concerning global weak solutions [ref] , local and global (small) strong solutions [ref] , singularity formation in Hall-MHD [ref] , and the asymptotic behavior of weak and strong solutions [ref] .
On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system · Introduction and main results · 1.00
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Local well-posedness is studied in [ref] for the Hall-MHD system with fractional magnetic diffusion.
Long time behavior of solutions to the 3D Hall-magneto-hydrodynamics system with one diffusion · Introduction · 0.94
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Intensive studies in the last two decades involve existence of weak solutions [ref] , regularity and blow-up criterion [ref] , well-posedness [ref] , ill-posedness [ref] , singularity formation [ref] , asymptotic behavior of solutions [ref] , non-uniqueness of weak solutions [ref] , etc.
Dyadic models with intermittency dependence for the Hall MHD · Introduction · 0.99
What it cites
Cites as background 1 paper
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The Hall-MHD equations have been mathematically investigated in several works ( [ref] ).
Singularity formation for the incompressible Hall-MHD equations without resistivity · Introduction · 0.99
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