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 . We prove the analyticity of the Stokes semigroup in for . This follows from the analysis of the Stokes resolvent problem for data in , . 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.
"Estimates for the Navier–Stokes equations in the half-space for nonlocalized data." Anal. PDE 13 (4) 945 - 1010, 2020. https://doi.org/10.2140/apde.2020.13.945