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

초록

We provide a new ε-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 ε-condition of the vorticity without relying on the ε-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. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.

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