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

TL;DR

This paper proves that Leray-Hopf weak solutions to the 3D Navier-Stokes equations with certain initial data are regular by constructing a supercritical space and deriving energy estimates, advancing understanding of solution regularity.

Contribution

The paper introduces a novel supercritical space with an inverse logarithmic weight and uses it to establish regularity of weak solutions to the 3D Navier-Stokes equations.

Findings

Weak solutions belong to L1(0,1; H1/2(R^3))

High frequency energy estimates lead to regularity

Supercritical space construction is key to the proof

Abstract

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.