Global weak Besov solutions of the Navier-Stokes equations and applications

TL;DR

This paper introduces a new class of global weak solutions to the 3D Navier-Stokes equations within critical Besov spaces, demonstrating stability and applications to blow-up phenomena and self-similarity.

Contribution

It defines a novel notion of global weak solutions in critical Besov spaces and establishes their stability, with new splitting results and applications to key problems in fluid dynamics.

Findings

Solutions satisfy stability under weak-* convergence

Applications to blow-up criteria and minimal initial data

New splitting result in homogeneous Besov spaces

Abstract

We introduce a notion of global weak solution to the Navier-Stokes equations in three dimensions with initial values in the critical homogeneous Besov spaces $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}$, $p > 3$. These solutions satisfy a certain stability property with respect to the weak-$\ast$ convergence of initial conditions. To illustrate this property, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. Our proof relies on a new splitting result in homogeneous Besov spaces that may be of independent interest.