On the blow-up criterion for the Hall-MHD problem with partial dissipation in R 3 $\mathbb{R}^{3}$

Abstract

Abstract In this paper, we investigate the 3D incompressible Hall-magnetohydrodynamics with partial dissipation. Based on the results in (Du in Bound. Value Probl. 2022:6, 2022; Du and Liu in Acta Math. Sci. 42A:5, 2022; Fei and Xiang in J. Math. Phys. 56:051504, 2015), we establish an improved blow-up criterion for classical solutions. Furthermore, using the blow-up criterion, we also obtain the existence of the classical solutions only under the condition that the initial data ∥ V 0 ∥ H 1 + ∥ B 0 ∥ H 2 $\|V_{0}\|_{H^{1}}+\|B_{0}\|_{H^{2}}$ are sufficiently small.

Keywords

• Blow-up criterion
• Hall-MHD equations
• Partial dissipation
• Small initial data