A locally anisotropic regularity criterion for the Navier–Stokes equation in terms of vorticity

Evan Miller

doi:10.1090/bproc/74

Abstract

In this paper, we will prove a regularity criterion that guarantees solutions of the Navier–Stokes equation must remain smooth so long as the vorticity restricted to a plane remains bounded in the scale critical space L t 4 L x 2 L^4_t L^2_x , where the plane may vary in space and time as long as the gradient of the unit vector orthogonal to the plane remains bounded. This extends previous work by Chae and Choe that guaranteed that solutions of the Navier–Stokes equation must remain smooth as long as the vorticity restricted to a fixed plane remains bounded in a family of scale critical spaces. This regularity criterion also can be seen as interpolating between Chae and Choe’s regularity criterion in terms of two vorticity components and Beirão da Veiga and Berselli’s regularity criterion in terms of the gradient of vorticity direction. In physical terms, this regularity criterion is consistent with key aspects of the Kolmogorov theory of turbulence, because it requires that finite-time blowup for solutions of the Navier–Stokes equation must be fully three dimensional at all length scales.

Highlights

The Navier–Stokes equation is the fundamental equation of fluid mechanics
In his foundational work on the Navier–Stokes equation, Leray proved the global existence of weak solutions to the Navier–Stokes equation satisfying an energy inequality for arbitrary initial data u0 ∈ L2 [15]
It is easy to check that this regularity criterion is scale critical with respect to the rescaling in (1.16)

Summary

Introduction

The Navier–Stokes equation is the fundamental equation of fluid mechanics. The incompressible Navier–Stokes equation is given by (1.1). In his foundational work on the Navier–Stokes equation, Leray proved the global existence of weak solutions to the Navier–Stokes equation satisfying an energy inequality for arbitrary initial data u0 ∈ L2 [15]. ∂tu − Δu = −(u · ∇)u − ∇p, in the sense of convolution with the heat kernel as in Duhamel’s formula They used this notion of solution, in particular the higher regularity that can be extracted from the heat kernel, to show that the Navier–Stokes equation has unique, smooth solutions locally in time for arbitrary initial data u0 ∈ H 1. Mild solutions are only known to exist locally in time, and so this approach based on the heat semigroup cannot guarantee the existence of global-in-time smooth solutions of the Navier–Stokes equation.

L2 dτ u0

H 1 exp

Proof of the main theorem

Relationship to the previous literature