Mild solutions and spacetime integral bounds for Stokes and Navier-Stokes flows in Wiener amalgam spaces

TL;DR

This paper establishes decay estimates and spacetime bounds for Stokes flows in Wiener amalgam spaces, constructs mild solutions for Navier-Stokes equations with small data, and explores stability and global solutions in these function spaces.

Contribution

It introduces new decay and spacetime bounds for Stokes flows in Wiener amalgam spaces and constructs global mild solutions for Navier-Stokes equations with small initial data.

Findings

Decay estimates for Stokes flows in amalgam spaces

Construction of global mild solutions for small data

Stability results for weak solutions with small local $L^3$ data

Abstract

We first prove decay estimates and spacetime integral bounds for Stokes flows in amalgam spaces $E^r_q$ which connect the classical Lebesgue spaces to the spaces of uniformly locally $r$-integrable functions. Using these estimates, we construct mild solutions of the Navier-Stokes equations in the amalgam spaces satisfying the corresponding spacetime integral bounds. Time-global solutions are constructed for small data in $E^3_q$, $1\le q \le 3$. Our results provide new bounds for the strong solutions classically constructed by Kato and the more recent solutions in uniformly local spaces constructed by Maekawa and Terasawa. As an application we obtain a result on the stability of suitability for weak solutions to the perturbed Navier-Stokes equation where the drift velocity solves the Navier-Stokes equations and has small data in a local $L^3$ class. Extending an earlier result, we also construct global-in-time local energy weak solutions in $E^2_q$, $1\le q <2$.