On strong solutions to the compressible Hall-magnetohydrodynamic system

Global existence and temporal decay for the 3D compressible Hall-magnetohydrodynamic system

“…When ρ = const, system (1.1) becomes the incompressible Hall-MHD system, which has received many studies, see [1,4,5,6,13,14,17,36,37,38]. When divu = 0, system (1.1) becomes the density-dependent Hall-MHD system, which has been investigated by many authors, and for more details, see [12,18].…”

“…However, to the best of our knowledge, very few results have been established on the dynamics of the global solutions to the 3D compressible Hall-MHD system, especially on the temporal decay of the solutions. Very recently, Fan et al [13] first obtained the global existence and the optimal decay rates for the 3D compressible Hall-MHD equations (1.1) where the initial data are close to an equilibrium state and belong to…”

“…In these known results mentioned above (See, [13,24,28]), L 1 (R 3 x ) integrability plays an important role in the proof of the optimal decay rates based on the spectral analysis of the semigroup. In general, for evolution equations in which L 2 (R 3 x ) based norms can be propagated by the solution, it is common to make a bounded assumption on the L 1 (R 3 x ) norm of the initial data and combine this with L 2 (R 3 x ) type estimates in order to obtain large time decay estimates.…”

“…Since the local strong solutions can be proven by [13], global solutions will follow in a standard continuity argument after we establish a priori estimate (1.5). We assume that that…”

Liouville‐type theorem for the steady compressible Hall‐MHD system

“…The time‐dependent compressible Hall‐MHD has been studied by many authors. See other studies for the study of global existence and temporal decay, and see Fan et al for the study of blow‐up criterion of strong solutions. In Mu, the zero Mach limit of the compressible Hall‐MHD system is studied.…”