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Well-posedness for the Navier-Stokes equations with datum in the Sobolev spaces. (English) Zbl 1375.35311
Summary: 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 [M. Cannone, Ondelettes, paraproduits, et Navier-Stokes. Paris: Diderot (1995; Zbl 1049.35517); M. Cannone and Y. Meyer, Methods Appl. Anal. 2, No. 3, 307–319 (1995; Zbl 0842.35074)]). 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, Rev. Mat. Iberoam. 13, No. 3, 515–541 (1997; Zbl 0897.35061)] in which \(p=d\) and \(s=0\).

MSC:
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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