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In this paper, we investigate the vanishing viscosity and magnetic diffusivity limits for the compressible magnetohydrodynamics (MHD) equations on a smooth bounded domain in R³. Under the assumptions of interior regularity and uniform local boundedness near the boundary, we establish a Kato-type criterion under which finite energy weak solutions to the dissipative MHD system converge to weak solutions to the ideal MHD system as the viscosity and magnetic resistivity coefficients decrease to zero. In particular, under such criterion, the anomalous energy dissipation in the vanishing viscosity and diffusivity limits for MHD is precluded.
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Li et al. (2026) studied this question.
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