Weighted estimates for the Stokes semigroup in the half-space

TL;DR

This paper develops new weighted estimates for the Stokes semigroup in the half-space, establishing existence, uniqueness, and regularity of solutions in weighted Lebesgue spaces, and extends the theory towards Navier-Stokes applications.

Contribution

It introduces a novel weight function and generalizes previous results, advancing the analysis of the Stokes system in weighted spaces for half-space and exterior domains.

Findings

Established existence and uniqueness of solutions in weighted spaces

Extended regularity results for the Stokes system

Paved the way for Navier-Stokes analysis in weighted settings

Abstract

We investigate the initial-boundary value problem for the Stokes system in the half-space, within the framework of weighted Lebesgue spaces. Introducing a new weight function defined via a product of powers of distances from fixed points, we establish existence, uniqueness, and regularity results for strong solutions to the Stokes problem in the half space. Our analysis generalizes previous results for the Stokes system in radial-weighted spaces (Galdi and Maremonti, J. Math. Fluid Mech. 25:7,2023; Maremonti and Pane, J. Math. Fluid Mech. 27:2,2025) and extends the theory to our setting. These results represent a first step toward the analysis of the Navier-Stokes system in weighted spaces, with applications in both half-space and exterior domain configurations.