Global Weak Solutions of the Navier–Stokes Equations for Intermittent Initial Data in Half-Space

We prove the existence of global-in-time weak solutions of the incompressible Navier–Stokes equations in the half-space R + 3 with initial data in a weighted space that allows non-uniformly locally square integrable functions that grow at large scales 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 a 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 ≥ ϵ 0 | x | 2 + M , where ϵ 0 > 0 is arbitrarily small and M > 0 . By adding conditions on the data within a weighted L 2 framework, we improve the algebraic bounds on the size of this region and we refine the pointwise decay rate of the solution within this region. 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.