# zbMATH — the first resource for mathematics

A note on local interior regularity of a suitable weak solution to the Navier-Stokes problem. (English) Zbl 1260.35125
Summary: We formulate a criterion which guarantees a local regularity of a suitable weak solution $$v$$ to the Navier-Stokes equations (in the sense of L. Caffarelli, R. Kohn and L. Nirenberg [Commun. Pure Appl. Math. 35, 771–831 (1982; Zbl 0509.35067)]). The criterion shows that if $$(x_0, t_0)$$ is a singular point of solution $$v$$ then the $$L^3$$-norm of $$v$$ concentrates in an amount greater than or equal to some $$\epsilon > 0$$ in an arbitrarily small neighbourhood of $$x_0$$ at all times $$t$$ in some left neighbourhood of $$t_0$$. As a partial result, we prove that a localized solution satisfies the strong energy inequality.

##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids
##### Keywords:
Navier-Stokes equations; suitable weak solution; regularity
Zbl 0509.35067
Full Text: