Global regularity of Leray-Hopf weak solutions to 3D Navier-Stokes equations

We show that any Leray-Hopf weak solution to 3D Navier-Stokes equations with initial values u0 2 H1=2(R3) belong to L1(0; 1; H1=2(R3)) and thus it is regular. For the proof, flrst, we construct a supercritical space, the norm of which is compared to the homogeneous Sobolev H_ 1=2-norm in that it has inverse logarithmic weight very sparsely in the frequency domain. Then we obtain the energy estimates of high frequency parts of the solution which involve the supercritical norm on the right-hand side. Finally, we superpose the energy norm of high frequency parts of the solution to get estimates of the critical norms of the weak solution via the re-scaling argument.