Acta Mathematica Vietnamica

, Volume 42, Issue 3, pp 431–443

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

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 

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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