Weighted estimates for the Stokes semigroup in the half-space
Angelica Pia Di Feola, Vittorio Pane
TL;DR
This work develops a weighted-Lebesgue framework for the Stokes system in the half-space by introducing a scaling-invariant weight $w(x)$ built from distances to fixed points. It establishes existence, uniqueness, and regularity of strong solutions, along with sharp time–space estimates and integral representations, extending the Stokes Cauchy theory to weighted spaces and transfering these tools to the half-space IBVP via extensions and Calderón–Zygmund theory. A key component is the weighted Helmholtz decomposition in $L^q_w(\mathbb{R}^n_+)$, proven in the Appendix, which underpins the decomposition of vector fields and the analysis of the Stokes operator. Collectively, the results provide foundational steps toward the Navier–Stokes system in weighted spaces on half-spaces and exterior domains, with potential applications to long-range behavior and asymptotics in such geometries.
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.