On the Initial Value Problem for the Navier-Stokes Equations with the Initial Datum in Critical Sobolev and Besov Spaces
Vol. 23 (2016), No. 2, Page 499--528.
Khai, D. Q. ; Tri, N. M.
On the Initial Value Problem for the Navier-Stokes Equations with the Initial Datum in Critical Sobolev and Besov Spaces
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Abstract:
The existence of local unique mild solutions to the Navier-Stokes equations in the whole space with an initial tempered distribution datum in critical homogeneous or inhomogeneous Sobolev spaces is shown. Especially, the case when the integral-exponent is less than 2 is investigated. The global existence is also obtained for the initial datum in critical homogeneous Sobolev spaces with a norm small enough in suitable critical Besov spaces. The key lemma is to establish the bilinear estimates in these spaces, due to the point-wise decay of the kernel of the heat semigroup.
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.
Mathematical Reviews Number: MR3469007
Received: 2014-10-27
