### Abstract

We show existence theorem of global mild solutions with small initial data and external forces in Lorentz spaces with scaling invariant norms. If the initial data have more regularity in another scaling invariant class, then our mild solution is actually the strong solution. The result on local existence of solutions for large data is also discussed. Our method is based on the maximal regularity theorem on the Stokes equations in Lorentz spaces. Then we apply our theorem to prove existence of self-similar solutions provided both initial data and external forces are homogeneous functions. Since we construct the global solution by means of the implicit function theorem, as a byproduct, its stability with respect to the given data is necessarily obtained.

Original language | English |
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Pages (from-to) | 1693-1708 |

Number of pages | 16 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 458 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2018 Feb 15 |

Externally published | Yes |

### Keywords

- Global solutions
- Implicit function theorem
- Lorentz space
- Maximal regularity theorem
- Navier–Stokes equations
- Self-similar solutions

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics