TY - JOUR
T1 - Remarks on the global regularity issue of the two-and-a-half-dimensional Hall-magnetohydrodynamics system
AU - Rahman, Mohammad Mahabubur
AU - Yamazaki, Kazuo
N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2022/10
Y1 - 2022/10
N2 - Whether or not the solution to the 212-dimensional Hall-magnetohydrodynamics system starting from any smooth initial data preserves its regularity for all time remains a challenging open problem. Although the research direction on component reduction of regularity criteria for Navier–Stokes equations and magnetohydrodynamics system has caught much attention recently, the Hall term has presented many difficulties. In this manuscript, we discover a certain cancellation within the Hall term and obtain various new regularity criteria: first, in terms of a gradient of only the third component of the magnetic field; second, in terms of only the third component of the current density; third, in terms of only the third component of the velocity field; fourth, in terms of only the first and second components of the velocity field. As another consequence of the cancellation that we discovered, we are able to prove the global well-posedness of the 212-dimensional Hall-magnetohydrodynamics system with hyper-diffusion only for the magnetic field in the horizontal direction; we also obtained an analogous result in the three-dimensional case via the discovery of additional cancellations. These results extend and improve various previous works.
AB - Whether or not the solution to the 212-dimensional Hall-magnetohydrodynamics system starting from any smooth initial data preserves its regularity for all time remains a challenging open problem. Although the research direction on component reduction of regularity criteria for Navier–Stokes equations and magnetohydrodynamics system has caught much attention recently, the Hall term has presented many difficulties. In this manuscript, we discover a certain cancellation within the Hall term and obtain various new regularity criteria: first, in terms of a gradient of only the third component of the magnetic field; second, in terms of only the third component of the current density; third, in terms of only the third component of the velocity field; fourth, in terms of only the first and second components of the velocity field. As another consequence of the cancellation that we discovered, we are able to prove the global well-posedness of the 212-dimensional Hall-magnetohydrodynamics system with hyper-diffusion only for the magnetic field in the horizontal direction; we also obtained an analogous result in the three-dimensional case via the discovery of additional cancellations. These results extend and improve various previous works.
KW - Global well-posedness
KW - Hall-magnetohydrodynamics system
KW - Magnetohydrodynamics system
KW - Navier–Stokes equations
KW - Regularity criteria
UR - http://www.scopus.com/inward/record.url?scp=85139218517&partnerID=8YFLogxK
U2 - 10.1007/s00033-022-01853-2
DO - 10.1007/s00033-022-01853-2
M3 - Article
AN - SCOPUS:85139218517
SN - 0044-2275
VL - 73
JO - Zeitschrift fur Angewandte Mathematik und Physik
JF - Zeitschrift fur Angewandte Mathematik und Physik
IS - 5
M1 - 217
ER -