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We prove existence of global-in-time weak solutions of the incompressibleNavier-Stokes equations in the half-space $\mathbb{R}^3_+$ with initial data ina weighted space that allow non-uniformly locally square integrable functionsthat grow at spatial infinity in an intermittent sense. The space for initialdata is built on cubes whose sides $R$ are proportional to the distance to theorigin 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 inthe weighted space, as well as new pressure estimates. Also, we prove eventualregularity 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, weconstruct global discretely self-similar solutions, thus extending the theoryon the half-space to the same generality as the whole space.

Global weak solutions of the Navier-Stokes equations for intermittent initial data in half-space