On the Cauchy problem of 3D nonhomogeneous magnetohydrodynamic equations with density-dependent viscosity and vacuum

Abstract

This paper concerns the Cauchy problem of the three-dimensional nonhomogeneous incompressible magnetohydrodynamic (MHD) equations with density-dependent viscosity and vacuum. We first establish some key a priori algebraic decay-in-time rates of the strong solutions. Then after using these estimates, we also obtain the global existence and large time asymptotic behavior of strong solutions in the whole three-dimensional space, provided that the initial velocity and magnetic field are suitable small in the \(\dot{H}^{\beta }\)-norm for some \(\beta \in (1/2, 1]\). Note that any smallness and compatibility conditions assumed on the initial data are not used in this result. Moreover, the density can contain vacuum states and even have compact support initially.

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Acknowledgements

The author would like to thank the anonymous referee for his/her helpful comments, which improve the presentation of the paper. This work is partially supported by Doctoral Research Foundation of Weifang University (Grant No. 2020BS05).

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Correspondence to Mingyu Zhang.

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Zhang, M. On the Cauchy problem of 3D nonhomogeneous magnetohydrodynamic equations with density-dependent viscosity and vacuum. Z. Angew. Math. Phys. 71, 106 (2020). https://doi.org/10.1007/s00033-020-01333-5

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Keywords

  • Incompressible magnetohydrodynamic equations
  • Density-dependent viscosity
  • Vacuum
  • Global strong solutions

Mathematics Subject Classification

  • 35Q35
  • 76D03
  • 76W05