## Abstract

This is a survey of recent results concerning on maximal L^{1}-regularity of the heat equation with the Naumann boundary condition in the half Euclidian space Ogawa and Shimizu (Proc Jpn Acad A, 96:57–62, 2020). It also includes maximal L^{1}-regularity for the initial boundary value of the Stokes system in the half-space under the stress free boundary condition. As an application, we introduce the time global well-posedness for the free boundary problem of the incompressible Navier-Stokes equations under the small initial data in the half Euclidean spaces R+n developed in Danchin-Hieber-Mucha-Tolksdorf (arXiv:2011.07918) and Ogawa and Shimizu (2021).

Original language | English |
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Pages (from-to) | 509-535 |

Number of pages | 27 |

Journal | Journal of Elliptic and Parabolic Equations |

Volume | 7 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2021 Dec |

## Keywords

- End-point estimate
- Free boundary problems
- Heat equations
- Initial-boundary value problems
- Maximal L-regularity
- The Neumann boundary condition

## ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Applied Mathematics

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