When the vaccum is allowed, if the global existence and uniqueness of strong solutions to three dimensional incompressible inhomogeneous magnetohydrodynamic equations holds true or not has always been a challenging open problem, even for the magnetofluids with special structures. In this paper, through deeply exploring the internal structure and characteristic of axisymmetric flows, we obtain some new discoveries and give a partial answer to above issue. More precisely, we prove that the axisymmetric magnetofluids flowing in the exterior of a cylinder will definitely admits a unique strong solution that exists globally in time without any compatibility conditions and small assumptions imposed on the initial data. Furthermore, we establish the algebraic decay rates for the time and spatial derivatives of both velocity and magnetic fields. To the best of our knowledge, this result gives the first unique 3D large solution existing globally in time.
Liu et al. (Mon,) studied this question.
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