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Acta Mathematica Vietnamica

, Volume 42, Issue 3, pp 431–443 | Cite as

Well-Posedness for the Navier-Stokes Equations with Datum in the Sobolev Spaces

  • Dao Quang Khai
Article

Abstract

In this paper, we study local well-posedness for the Navier-Stokes equations with arbitrary initial data in homogeneous Sobolev spaces \(\dot {H}^{s}_{p}(\mathbb {R}^{d})\) for \(d \geq 2, p > \frac {d}{2}\), and \(\frac {d}{p} - 1 \leq s < \frac {d}{2p}\). The obtained result improves the known ones for p > d and s = 0 (see [4, 6]). In the case of critical indexes \(s=\frac {d}{p}-1\), we prove global well-posedness for Navier-Stokes equations when the norm of the initial value is small enough. This result is a generalization of the one in [5] in which p = d and s = 0.

Keywords

Navier-Stokes equations Existence and uniqueness of local and global mild solutions Critical Sobolev and Besov spaces 

Mathematics Subject Classification (2010)

Primary 35Q30 Secondary 76D05 76N10 

Notes

Acknowledgments

This research was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2014.50.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Institute of Mathematics, Vietnam Academy of Science and TechnologyCau GiayVietnam

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