On local well-posedness of 3D ideal Hall-MHD system with an azimuthal magnetic field
arxiv(2024)
摘要
In this paper, we study the local well-posedness of classical solutions to
the ideal Hall-MHD equations whose magnetic field is supposed to be azimuthal
in the L^2-based Sobolev spaces. By introducing a good unknown coupling with
the original unknowns, we overcome difficulties arising from the lack of
magnetic resistance, and establish a self-closed H^m with (3≤
m∈ℕ) local energy estimate of the system. Here, a key cancellation
related to θ derivatives is discovered. In order to apply this
cancellation, part of the high-order energy estimates is performed in the
cylindrical coordinate system, even though our solution is not assumed to be
axially symmetric.
During the proof, high-order derivative tensors of unknowns in the
cylindrical coordinates system are carefully calculated, which would be useful
in further researches on related topics.
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