## Abstract

We study the Serrin-type regularity criteria for the solutions to the four-dimensional Navier-Stokes equations and magnetohydrodynamics system. We show that the sufficient condition for the solution to the four-dimensional Navier-Stokes equations to preserve its initial regularity for all time may be reduced in the following ways: from a bound on the four-dimensional velocity vector field to any two of its four components; from a bound on the gradient of the velocity vector field to the gradient of any two of its four components; and from a gradient of the pressure scalar field to any two of its partial derivatives. Results are further generalized to the magnetohydrodynamics system. These results may be seen as a four-dimensional extension of many analogous results that exist in the three-dimensional case and also component reduction results of many classical results.

Original language | English |
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Pages (from-to) | 2229-2252 |

Number of pages | 24 |

Journal | Communications in Mathematical Sciences |

Volume | 14 |

Issue number | 8 |

DOIs | |

State | Published - 2016 |

## Keywords

- Magnetohydrodynamics system
- Navier-Stokes equations
- Regularity criteria
- Scaling-invariance