  We prove existence of global-in-time weak solutions of the incompressible
Navier-Stokes equations in the half-space $\mathbb{R}^3_+$ with initial data in
a weighted space that allow non-uniformly locally square integrable functions
that grow at spatial infinity in an intermittent sense. The space for initial
data is built on cubes whose sides $R$ are proportional to the distance to the
origin and the square integral of the data is allowed to grow as a power of
$R$.
  The existence is obtained via a new a priori estimate and stability result in
the weighted space, as well as new pressure estimates. Also, we prove eventual
regularity of such weak solutions, up to the boundary, for $(x,t)$ satisfying
$t>c_1|x|^2 + c_2$, where $c_1,c_2>0$, for a large class of initial data $u_0$,
with $c_1$ arbitrarily small. As an application of the existence theorem, we
construct global discretely self-similar solutions, thus extending the theory
on the half-space to the same generality as the whole space.
