### Abstract

We study the local and global well-posedness of a full system of magnetohydrodynamic equations. The system is a coupling of the incompressible Navier - Stokes equations with the Maxwell equations through the Lorentz force and Ohm's law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale-invariant spaces classically used for Navier - Stokes. These solutions are global if the initial data are small enough. Our results not only simplify and unify the proofs for the space dimensions 2 and 3, but also refine those in [8]. The main simplification comes from an a priori L t ^{2} (L x ^{∞}) estimate for solutions of the forced Navier - Stokes equations.

Original language | English (US) |
---|---|

Pages (from-to) | 71-86 |

Number of pages | 16 |

Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

Volume | 144 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2014 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)