Estimates for the navier-stokes equations in the half-space for nonlocalized data

This paper is devoted to the study of the Stokes and Navier-Stokes equations, in a half-space, for initial data in a class of locally uniform Lebesgue integrable functions, namely L^q _{uloc, σ} (R^d ₊). We prove the analyticity of the Stokes semigroup e^{-t A} in L^q _{uloc, σ} (R^d ₊) for 1 < q ≤∞. This follows from the analysis of the Stokes resolvent problem for data in L^q _{uloc, σ} (R^d ₊), 1<q ≤∞. We then prove bilinear estimates for the Oseen kernel, which enables us to prove the existence of mild solutions. The three main original aspects of our contribution are: the proof of Liouville theorems for the resolvent problem and the time-dependent Stokes system under weak integrability conditions, the proof of pressure estimates in the half-space, and the proof of a concentration result for blow-up solutions of the Navier-Stokes equations. This concentration result improves a recent result by Li, Ozawa and Wang and provides a new proof.