Weak Solutions for the Navier-Stokes Equations for ${B}^{-1(ln)}_{\infty\infty}+{B}_{\dot{X}_r}^{-1+r,\frac{2}{1-r}}+L^2$ Initial Data

TL;DR

This paper extends the class of initial data spaces for which global weak solutions to the Navier-Stokes equations exist, broadening previous results to include more generalized Besov and logarithmic spaces.

Contribution

It introduces a larger initial data space combining Besov, logarithmic, and $L^2$ spaces, and proves the existence of weak solutions for Navier-Stokes equations within this space.

Findings

Established existence of weak solutions for initial data in the new combined space.

Extended previous results to include logarithmic Besov spaces.

Broadened the understanding of initial conditions leading to weak solutions.

Abstract

In 1934 Leray proved that the Navier-Stokes equations have global weak solutions for initial data in $L^2(\mathbb{R}^N)$. In 1990 Calder\'on extended this result to the initial value spaces $L^p(\mathbb{R}^N)$ ($2\leq p<\infty$). In the book "{\em Recent developments in the Navier-Stokes problems}" (2002), Lemari\'e-Rieusset extended this result of Calder\'on to the space $B_{\widetilde{X}_r}^{-1+r,\frac{2}{1-r}}(\mathbb{R}^N)+L^2(\mathbb{R}^N)$ ($0<r<1$), where ${X}_r$ is the space of functions whose pointwise products with $H^r$ functions belong to $L^2$, $\widetilde{X}_r$ denotes the closure of $C_0^\infty(\mathbb{R}^N)$ in ${X}_r$, and $B_{\widetilde{X}_r}^{-1+r,\frac{2}{1-r}}(\mathbb{R}^N)$ is the Besov space over $\widetilde{X}_r$. In this paper we further extend this result of Lemari\'e-Rieusset to the larger initial value space ${B}^{-1(ln)}_{\infty\infty}(\mathbb{R}^N)+{B}_{\widetilde{\dot{X}}_r}^{-1+r,\frac{2}{1-r}}(\mathbb{R}^N)+L^2(\mathbb{R}^N)$ ($0<r<1$).