Energy Superposition and Regularity for 3D Navier-Stokes Equations in the Largest Critical Space

Abstract

We show that a Leray-Hopf weak solution to the 3D Navier-Stokes Cauchy problem belonging to the space $L^\infty(0,T; B^{-1}_{\infty,\infty}(\mathbb R^3))$ is regular in $(0,T]$. As a consequence, it follows that any Leray-Hopf weak solution to the 3D Navier-Stokes equations is regular while it is temporally bounded in the largest critical space $\dot{B}^{-1}_{\infty,\infty}(\mathbb R^3)$ as well as in any critical spaces. For the proof we present a new elementary method which is to superpose the energy norm of high frequency parts in an appropriate way to generate higher order norms. Thus, starting from the energy estimates of high frequency parts of a weak solution, one can obtain its estimates of higher order norms. By a linear energy superposition we get very simple and short proofs for known regularity criteria for Leray-Hopf weak solutions in endpoint Besov spaces $B^{\sigma}_{\infty,\infty}$ for $\sigma\in [-1,0)$, the extension of Prodi-Serrin conditions. The main result of the paper is proved by applying technique of a nonlinear energy superposition and linear energy superpositions, repeatedly. The energy superposition method developed in the paper can also be applied to other supercritical nonlinear PDEs.