### Abstract

In this paper, we consider the global well-posedness of a three-dimensional incompressible MHD type system with smooth initial data that is close to some nontrivial steady state. It is a coupled system between the Navier-Stokes equations and a free transport equation with a universal nonlinear coupling structure. The main difficulty of the proof lies in exploring the dissipative mechanism of the system due to the fact that there is a free transport equation of φ{symbol} in the coupled equations and only the horizontal derivatives of φ{symbol} is dissipative with respect to time. To achieve this, we first employ anisotropic Littlewood-Paley analysis to establish the key L^{1}(ℝ^{+}; Lip(ℝ^{3})) estimate to the third component of the velocity field. Then we prove the global well-posedness to this system by the energy method, which depends crucially on the divergence-free condition of the velocity field.

Original language | English (US) |
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Pages (from-to) | 531-580 |

Number of pages | 50 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 67 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2014 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Communications on Pure and Applied Mathematics*,

*67*(4), 531-580. https://doi.org/10.1002/cpa.21506