The Navier-Stokes Cauchy problem in a class of weighted function spaces

TL;DR

This paper extends the analysis of the Navier-Stokes equations to n-dimensional spaces with weighted initial data, proving local existence and decay properties of solutions in a generalized setting.

Contribution

It generalizes previous 3D stability results to nD, establishing local existence and decay of solutions with weighted initial data.

Findings

Proved local existence of unique regular solutions in nD.

Established spatial decay rates linked to weight functions.

Extended stability analysis to higher dimensions.

Abstract

We consider the Navier-Stokes Cauchy problem with an initial datum in a weighted Lebesgue space. The weight is a radial function increasing at infinity. Our study partially follows the ideas of the paper by G.P. Galdi and P. Maremonti "On the stability of steady-state solutions to the Navier-Stokes equations in the whole space", JMFM, 25 (2023). The authors of the quoted paper consider a spatial study of stability of steady fluid motions. The result hold in 3D and for small data. Here, relatively to the perturbations of the rest state, we generalize the result. We study the nD Navier-Stokes Cauchy problem, n greater than 2. We prove the existence (local) of a unique regular solution. Moreover, the solution enjoys a spatial asymptotic decay whose order of decay is connected to the weight.