On global solutions to the Navier-Stokes system with large $L^{3,\infty}$ initial data

TL;DR

This paper investigates the behavior of global solutions to the Navier-Stokes equations with initial data in the weak Lebesgue space $L^{3,ty}$, extending previous results from the Lebesgue space $L_3$ to this larger, scale-invariant space.

Contribution

It extends the analysis of global Navier-Stokes solutions from $L_3$ to the more general, scale-invariant space $L^{3,ty}$, addressing the definition of weak solutions with such initial data.

Findings

Addresses the behavior of solutions with $L^{3,ty}$ initial data.

Provides insights into the formulation of global weak solutions in non-energy spaces.

Extends previous $L_3$ results to the weak Lebesgue space $L^{3,ty}$.

Abstract

This paper addresses a question concerning the behaviour of a sequence of global solutions to the Navier-Stokes equations, with the corresponding sequence of smooth initial data being bounded in the (non-energy class) weak Lebesgue space $L^{3,\infty}$. It is closely related to the question of what would be a reasonable definition of global weak solutions with a non-energy class of initial data, including the aforementioned Lorentz space. This paper can be regarded as an extension of a similar problem regarding the Lebesgue space $L_3$ to the weak Lebesgue space $L^{3,\infty}$, whose norms are both scale invariant with the respect to the Navier-Stokes scaling.