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

In this paper, we study local well-posedness for the Navier-Stokes equations with arbitrary initial data in homogeneous Sobolev spaces Ḣps(ℝd) for d≥2,p>d2, and dp−1≤s<d2p. The obtained result improves the known ones for p > d and s = 0 (see [4, 6]). In the case of critical indexes s=dp−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. © 2016, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore.