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 ${\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}$, $p>3$. 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.

arXiv commentShort version, 63 pages, 2 figures. Includes new appendix on splitting lemmas. Submitted

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• Spatial decay of discretely self-similar solutions to the Navier-Stokes equations
• Scale-invariant estimates and vorticity alignment for Navier-Stokes in the half-space with no-slip boundary conditions
• Global weak solutions of the Navier-Stokes equations for intermittent initial data in half-space
• Global regularity for solutions of the Navier-Stokes equation sufficiently close to being eigenfunctions of the Laplacian