### 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 |

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### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Navier–Stokes equations with external forces in Lorentz spaces and its application to the self-similar solutions.** / Kozono, Hideo; Shimizu, Senjo.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 458, no. 2, pp. 1693-1708. https://doi.org/10.1016/j.jmaa.2017.10.048

}

TY - JOUR

T1 - Navier–Stokes equations with external forces in Lorentz spaces and its application to the self-similar solutions

AU - Kozono, Hideo

AU - Shimizu, Senjo

PY - 2018/2/15

Y1 - 2018/2/15

N2 - 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.

AB - 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.

KW - Global solutions

KW - Implicit function theorem

KW - Lorentz space

KW - Maximal regularity theorem

KW - Navier–Stokes equations

KW - Self-similar solutions

UR - http://www.scopus.com/inward/record.url?scp=85034208929&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85034208929&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2017.10.048

DO - 10.1016/j.jmaa.2017.10.048

M3 - Article

AN - SCOPUS:85034208929

VL - 458

SP - 1693

EP - 1708

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -