Well-posedness for the Navier-Stokes equations with datum in the Sobolev spaces

Abstract

In this paper, we study local well-posedness for the Navier-Stokes \linebreak equations with arbitrary initial data in homogeneous Sobolev spaces ${\dot{H}}_p^s({\mathbb{R}}^d)$ for $d\ge 2,p>\frac{d}{2},\mathrm{and}\frac{d}{p}-1\le s<\frac{d}{2p}$. The obtained result improves the known ones for $p>d$ and $s=0$ M. Cannone and Y. Meyer (1995). 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 M. Cannone (1997) in which $p=d$ and $s=0$.

arXiv comment14 pages. arXiv admin note: substantial text overlap with arXiv:1603.04219, arXiv:1601.01441

References in corpus (1)

• Well-posedness for the Navier-Stokes equations with datum in Sobolev-Fourier-Lorentz spaces

Cited by in corpus (1)

• Well-posedness for the Navier-Stokes equations with data in homogeneous Sobolev-Lorentz spaces