Triebel-Lizorkin-Lorentz spaces and the Navier-Stokes equations

TL;DR

This paper develops foundational properties of Triebel-Lizorkin-Lorentz spaces and applies them to establish local well-posedness results for the Navier-Stokes equations within these function spaces.

Contribution

It introduces and analyzes Triebel-Lizorkin-Lorentz spaces, proving key properties and applying them to PDEs, notably the Navier-Stokes equations.

Findings

Triebel-Lizorkin-Lorentz spaces are of class $\\mathcal{HT}$

They have property $(\alpha)$ and Mikhlin multiplier results

Laplace and Stokes operators admit bounded $H^\infty$-calculus in these spaces

Abstract

We derive basic properties of Triebel-Lizorkin-Lorentz spaces important in the treatment of PDE. For instance, we prove Triebel-Lizorkin-Lorentz spaces to be of class $\mathcal{HT}$, to have property $(\alpha)$, and to admit a multiplier result of Mikhlin type. By utilizing these properties we prove the Laplace and the Stokes operator to admit a bounded $H^\infty$-calculus. This is finally applied to derive local strong well-posedness for the Navier-Stokes equations on corresponding Triebel-Lizorkin-Lorentz ground spaces.