Two Hall-MHD solutions that share their low Fourier modes up to a time-dependent determining wavenumber converge to each other in L2 as time goes to infinity.
Local well-posedness for the Hall-MHD system in optimal Sobolev Spaces
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abstract
We show that the viscous resistive magneto-hydrodynamics system with Hall effect is locally well-posed in $H^s(\mathbb R^n)\times H^{s+1-\varepsilon}(\mathbb R^n)$ with $s>\frac{n}2-1$ and any small enough $\varepsilon>0$ such that $s+1-\varepsilon>\frac{n}2$. This space is to date the largest local well-posedness space in the class of Sobolev spaces for the system. It is also optimal according to the predominant scalings of the two equations in the system.
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Determining wavenumbers for the incompressible Hall-magneto-hydrodynamics
Two Hall-MHD solutions that share their low Fourier modes up to a time-dependent determining wavenumber converge to each other in L2 as time goes to infinity.