A New Condition on the Vorticity for Partial Regularity of a Local Suitable Weak Solution to the Navier–Stokes Equations

Abstract

We provide a new \(\varepsilon \)-condition for the vorticity of a suitable weak solution to the Navier–Stokes equations that leads to partial regularity. This refines the well known limsup condition of the Caffarelli-Kohn-Nirenberg Theorem by a new condition on the vorticity, replacing limsup by a suitable range of the radius r of the parabolic cylinders. As a consequence, the partial regularity is obtained directly from this \(\varepsilon \)-condition of the vorticity without relying on the \(\varepsilon \)-condition of the velocity. Furthermore, by the local nature of the method this result holds for any local suitable weak solution of the Navier–Stokes equations in a general domain.