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
.