On estimates for the Stokes flow in a space of bounded functions

TL;DR

This paper investigates the regularizing effects of certain operators related to the Stokes semigroup and Helmholtz projection, establishing new estimates that ensure unique solutions of Navier-Stokes equations in bounded function spaces.

Contribution

It provides novel $L^{ abla}$-estimates for the Stokes operator in bounded function spaces, enabling existence results for Navier-Stokes solutions.

Findings

New a priori $L^{ abla}$-estimates for the Stokes operator

Existence of mild solutions for Navier-Stokes in bounded functions

Applicable to bounded and exterior domains

Abstract

In this paper, we study regularizing effects of the composition operator $S(t)\mathbb{P}\partial$ for the Stokes semigroup $S(t)$ and the Helmholtz projection $\mathbb{P}$ in a space of bounded functions. We establish new a priori $L^{\infty}$-estimates of the operator $S(t)\mathbb{P}\partial$ for a certain class of domains including bounded and exterior domains. They imply unique existence of mild solutions of the Navier-Stokes equations in a space of bounded functions.