Properties of Navier-Stokes mild solutions with initial data in subcritical Lorentz spaces

TL;DR

This paper investigates the properties of mild solutions to the Navier-Stokes equations with initial data in subcritical Lorentz spaces, establishing propagation of regularity and uniqueness results within this functional framework.

Contribution

It extends local existence, uniqueness, and regularity propagation results for Navier-Stokes solutions to the setting of Lorentz spaces, using Kato's method and Lorentz space techniques.

Findings

Solutions cannot blow up in Lorentz norm before the associated Besov norm

Provides a self-contained local theory in Lorentz spaces including blow-up estimates

Establishes continuity and uniqueness properties of solutions in Lorentz spaces

Abstract

For initial data $f$ in a subcritical Lorentz space $L^{p,q}(\mathbb{R}^{n}) \hookrightarrow \dot B^{-\frac np}_{\infty,\infty}(\mathbb{R}^n)$ ($n<p<\infty$, $1\leq q \leq \infty$), we prove results which imply in particular that a local in time mild Navier-Stokes solution cannot become unbounded in the $L^{p,q}(\mathbb{R}^{n})$-norm before it becomes unbounded in the norm of the larger subcritical Besov space $\dot B^{-\frac np}_{\infty,\infty}(\mathbb{R}^n)$. In view of the known local theory in such large Besov spaces, this can be thought of as a `propagation of regularity' type of result; here, we provide a self-contained local theory (including scaling-appropriate `blow-up estimates', similar to those established by J. Leray in the Lebesgue setting) in the Lorentz setting, along with uniqueness results which imply such propagation of regularity. Our existence results are based on the method of T. Kato (1984) with Lorentz spaces replacing Lebesgue spaces throughout, and are given without any reference to the Besov framework. The uniqueness results are similarly self-contained, extending certain Lebesgue space methods of Fabes-Jones-Riviere (1972) to the Lorentz setting. We also establish certain continuity properties of the solutions which are constructed. (A more detailed abstract is provided in the article itself.)