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

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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.

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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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Correspondence to Dao Quang Khai.

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Khai, D.Q. Well-Posedness for the Navier-Stokes Equations with Datum in the Sobolev Spaces. Acta Math Vietnam 42, 431–443 (2017). https://doi.org/10.1007/s40306-016-0192-x

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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