Caffarelli–Kohn–Nirenberg partial regularity theorem

Statement

Let $(u,p)$ be an unforced suitable weak solution of the three-dimensional Navier–Stokes equations on an open cylinder $Q$ , with $\nu>0$ . Its interior singular set satisfies

\mathcal H^1_{\rm par}(S(u))=0,

where $\mathcal H^1_{\rm par}$ is parabolic one-dimensional Hausdorff measure. This is the unforced interior form of the Caffarelli–Kohn–Nirenberg partial regularity theorem.

It is a partial regularity statement.

Scope

The conclusion is partial regularity, not emptiness of the singular set. Boundary versions and forced versions require additional hypotheses and estimates. The theorem uses suitability, including its local energy inequality.

References

Caffarelli, Kohn and Nirenberg, Partial regularity of suitable weak solutions (1982).