Well-posedness for the Navier-Stokes equations with data in homogeneous Sobolev-Lorentz spaces

TL;DR

This paper establishes local well-posedness for the Navier-Stokes equations with initial data in homogeneous Sobolev-Lorentz spaces, extending previous results and proving global well-posedness under small initial data in critical cases.

Contribution

It generalizes existing well-posedness results for Navier-Stokes equations to broader Sobolev-Lorentz spaces, including critical index cases, and proves global solutions for small initial data.

Findings

Local well-posedness in Sobolev-Lorentz spaces

Global well-posedness for small initial data at critical indexes

Extension of previous results to broader function spaces

Abstract

In this paper, we study local well-posedness for the Navier-Stokes equations (NSE) with the arbitrary initial value in homogeneous Sobolev-Lorentz spaces $\dot{H}^s_{L^{q, r}}(\mathbb{R}^d):= (-\Delta)^{-s/2}L^{q,r}$ for $d \geq 2, q > 1, s \geq 0$, $1 \leq r \leq \infty$, and $ \frac{d}{q}-1 \leq s < \frac{d}{q}$, this result improves the known results for $q > d,r=q, s = 0$ (see M. Cannone (1995) and M. Cannone and Y. Meyer (1995)) and for $q =r= 2, \frac{d}{2} - 1 < s < \frac{d}{2}$ (see M. Cannone (1995, J. M. Chemin (1992)). In the case of critical indexes ($s=\frac{d}{q}-1$), we prove global well-posedness for NSE provided the norm of the initial value is small enough. The result that is a generalization of the result of M. Cannone (1997) for $q = r=d, s=0$.