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Acta Mathematica Vietnamica

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

icon-email Dao Quang Khai

Abstract

In this paper, we study local well-posedness for the Navier-Stokes equations with arbitrary initial data in homogeneous Sobolev spaces H˙ps(Rd) for d2,p>d2, and dp1s<d2p. The obtained result improves the known ones for p>d and s=0 (see [4, 6]). In the case of critical indexes s=dp1, 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.