Another remark on the global regularity issue of the Hall-magnetohydrodynamics system

• Published in: Journal of evolution equations 2024-09, Vol.24 (3), Article 70
• Main Authors: Rahman, Mohammad Mahabubur, Yamazaki, Kazuo
• Format: Article
• Language: English
• Subjects: Analysis; Magnetic diffusion; Magnetohydrodynamics; Mathematics; Mathematics and Statistics; Regularity
• ISSN: 1424-3199; 1424-3202
• DOI: 10.1007/s00028-024-01000-6

We discover new cancellations upon H 2 ( R n ) -estimate of the Hall term, n ∈ { 2 , 3 } . Consequently, first, we derive a regularity criterion for the 3-dimensional Hall-magnetohydrodynamics system in terms of horizontal components of velocity and magnetic fields. Second, we are able to prove the global regularity of the 2 1 2 -dimensional electron magnetohydrodynamics system with magnetic diffusion ( - Δ ) 3 2 ( b 1 , b 2 , 0 ) + ( - Δ ) α ( 0 , 0 , b 3 ) for α > 1 2 despite the fact that ( - Δ ) 3 2 is the critical diffusive strength. Lastly, we extend this result to the 2 1 2 -dimensional Hall-magnetohydrodynamics system with - Δ u replaced by ( - Δ ) α ( u 1 , u 2 , 0 ) - Δ ( 0 , 0 , u 3 ) for α > 1 2 . The sum of the derivatives in diffusion that our result requires is 11 + ϵ for any ϵ > 0 , while the sum for the classical 2 1 2 -dimensional Hall-magnetohydrodynamics system is 12 considering - Δ u and - Δ b , of which its global regularity issue remains an outstanding open problem.