## Abstract

We prove the first ever pointwise estimates of the (unrestricted) Green tensor and the associated pressure tensor of the nonstationary Stokes system in the half-space, for every space dimension greater than one. The force field is not necessarily assumed to be solenoidal. The key is to find a suitable Green tensor formula which maximizes the tangential decay, showing in particular the integrability of Green tensor derivatives. With its pointwise estimates, we show the symmetry of the Green tensor, which in turn improves pointwise estimates. We also study how the solutions converge to the initial data, and the (infinitely many) restricted Green tensors acting on solenoidal vector fields. As applications, we give new proofs of existence of mild solutions of the Navier–Stokes equations in L^{q}, pointwise decay, and uniformly local L^{q} spaces in the half-space.

Original language | English |
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Journal | Communications in Mathematical Physics |

DOIs | |

Publication status | Accepted/In press - 2023 |

### Bibliographical note

Funding Information:The research of KK was partially supported by NRF-2019R1A2C1084685. The research of BL was partially supported by NSFC-11971148 and Hunan provincial NSF-2022jj10032. The research of both CL and TT was partially supported by the NSERC Grant RGPIN-2018-04137. The research of CL is supported in part by the Simons Foundation Math + X Investigator Award #376319 (Michael I. Weinstein).

Publisher Copyright:

© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics