Regularity aspects of the Navier-Stokes equations in critical spaces

Abstract

For better or for worse, our current understanding of the Navier-Stokes regularity problem is intimately connected with certain dimensionless quantities known as critical norms. In this thesis, we concern ourselves with one of the most basic questions about Navier-Stokes regularity: How must the critical norms behave at a potential Navier-Stokes singularity? In Chapter 2, we give a broad overview of the Navier-Stokes theory necessary to answer this question. This chapter is suitable for newcomers to the field. Next, we present two of our published papers [4,5] which answer this question in the context of homogeneous Besov spaces. In Chapter 3, we demonstrate that the critical Besov norms $\| u(\cdot,t) \|_{\dot B^{-1+3/p}_{p,q}(\R^3)}$, $p,q \in (3,+\infty)$, must tend to infinity at a potential singularity. Our proof has been streamlined from the published version [4]. In Chapter 4 (joint work with Tobias Barker), we develop a framework of global weak Besov solutions with initial data belonging to $\dot B^{-1+3/p}_{p,\infty}(\R^3)$, $p \in (3,+\infty)$. To illustrate this framework, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. This chapter has been reproduced from the published version [5].