A note on local interior regularity of a suitable weak solution to the Navier-Stokes problem. (English) Zbl 1260.35125

Neustupa, Jiří

Discrete Contin. Dyn. Syst., Ser. S 6, No. 5, 1391-1400 (2013).

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