The existence of weak solutions for the Navier-Stokes equations for the infinite cylinder with initial data in
L
p
{L^p}
is considered in this paper. We study the case of initial data in
L
p
(
R
n
)
{L^p}({R^n})
,
2
>
p
>
n
2 > p > n
, and
n
=
3
,
4
n = 3,4
. An existence theorem is proved covering these important cases and therefore, the "gap" between the Hopf-Leray theory
(
p
=
2
)
(p = 2)
and that of Fabes-Jones-Riviere
(
p
>
n
)
(p > n)
is bridged. The existence theorem gives a new method of constructing global solutions. The cases
p
=
n
p = n
are treated at the end of the paper.