ABSTRACT
This work establishes local existence and uniqueness as well as blow-up criteria for solutions of the Navier–Stokes equations in Sobolev–Gevrey spaces . More precisely, if it is assumed that the initial data belongs to , with , we prove that there is a time T>0 such that for and . If the maximal time interval of existence of solutions is finite, , then, we prove, for example, that the blow-up inequality holds for , a>0, ( is the integer part of ).
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Wilberclay G. Melo http://orcid.org/0000-0003-4388-3491
Notes
1. is the Sobolev–Gevrey space endowed with the inner product ( is the set of tempered distributions). Here and
2. is the homogenous Sobolev space.
3. The Sobolev–Gevrey space is endowed with the inner product .
4. In the Navier–Stokes equations (Equation1(1) (1) ), we have that where and and (i = 1, 2, 3).
5. The tensor product is given by where .