Global regularity for the Navier-Stokes equations with application to global solvability for the Euler equations

We show that any Leray-Hopf weak solution to the $d$-dimensional Navier-Stokes equations $(d\ge 3)$ with initial values $u_0\in H^s({\mathbb{R}}^d)$, $s\ge -1+\frac{d}{2}$, belongs to $L^{\mathrm{\infty }}(0,\mathrm{\infty };H^s({\mathbb{R}}^d))$ and thus it is globally regular. For the proof, first, we construct a supercritical space which has very sparse inverse logarithmic weight in the frequency domain, compared to the critical homogeneous Sobolev ${\dot{H}}^{-1+d\mathrm{/}2}$-norm. Then we obtain the energy estimates of high frequency parts of the solution which involve the supercritical norm as a factor of the upper bounds. Finally, we superpose the energy norm of high frequency parts of the solution to get estimates of the critical and subcritical norms independent of the viscosity coefficient for the weak solution via the re-scaling argument.