Global Weak Besov Solutions of the Navier–Stokes Equations and Applications

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.