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

TL;DR

The paper addresses global regularity for the d≥3 Navier-Stokes equations with initial data in H^s, s≥-1+d/2, by introducing a frequency-weighted, scaling-variant supercritical space X_1 and deriving high-frequency energy estimates that are uniform in viscosity. A re-scaling argument coupled with careful dyadic energy analysis yields viscosity-independent bounds, leading to global regularity and uniqueness for Leray-Hopf weak solutions and enabling vanishing-viscosity limits to establish global solvability for the Euler equations. The key contributions are the construction and utilization of X_1 to achieve subcritical control from a supercritical framework, and the resulting global-in-time estimates that bridge Navier-Stokes regularity with Euler solvability. This work provides a robust approach near the critical threshold and has significant implications for the long-time behavior of incompressible flows.

Abstract

We show that any Leray-Hopf weak solution to the $d$-dimensional Navier-Stokes equations $(d\geq 3)$ with initial values $u_0\in H^{s}(\mathbb R^d)$, $s\geq -1+\frac{d}{2}$, belongs to $L^\infty(0,\infty; H^{s}(\mathbb R^d))$ 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 $\dot{H}^{-1+d/2}$-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.

Theorems & Definitions (8)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1
• Lemma 3.1
• Lemma 3.2
• Remark 3.3
• Remark 4.1
• Theorem 4.2