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We are concerned with the study of the Cauchy problem to the 3D compressibleHall-magnetohydrodynamic system. We first establish the unique globalsolvability of strong solutions to the system when the initial data are closeto a stable equilibrium state in critical Besov spaces. Furthermore, under asuitable additional condition involving only the low frequencies of the dataand in $L^{2}$-critical regularity framework, we exhibit the optimal time decayrates for the constructed global solutions. The proof relies on an applicationof Fourier analysis to a mixed parabolic-hyperbolic system, and on a refinedtime-weighted energy functional.

Existence, uniqueness and optimal decay rates for the 3D compressible Hall-magnetohydrodynamic system