Abstract
This paper concerns the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum on \(\varOmega \subset \mathbb {R}^3\). The domain \(\varOmega \subset \mathbb {R}^3\) is a general connected smooth one, either bounded or unbounded. In particular, the initial density can have compact support when \(\varOmega \) is unbounded. First, we obtain the local existence and uniqueness of strong solution to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations without any compatibility condition assumed on the initial data. Then, we also prove the continuous dependence of strong solution on the initial data under an additional compatibility condition.
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Song, S. On local strong solutions to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum. Z. Angew. Math. Phys. 69, 23 (2018). https://doi.org/10.1007/s00033-018-0915-z
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DOI: https://doi.org/10.1007/s00033-018-0915-z