The IBVP for the Navier-Stokes equations in the half-space in a class of weighted Lebesgue spaces

Abstract

We study the initial-boundary value problem for the Navier-Stokes equations in the half-space with initial data belonging to a suitable weighted Lebesgue space. More precisely, we consider a weighted Lebesgue space associated with a weight function defined as a product of powers of the distances from finitely many fixed points. This framework was introduced in a previous work by A. P. Di Feola and V. Pane (J. Math. Anal. Appl. 558 (2026), 130390) for the study of the initial-boundary value problem for the Stokes system. The present paper completes that analysis and, at the same time, generalizes to the initial-boundary value problem the results obtained by Maremonti and Pane (J. Math. Fluid Mech. 27 (2025), Art. 2) for the Navier-Stokes Cauchy problem. We prove the existence (local) and uniqueness of a smooth solution and derive $L^q$-estimates, with $q>n$, together with the spatial asymptotic behavior of the velocity field.