A geometric improvement of the velocity-pressure local regularity criterion for a suitable weak solution to the Navier-Stokes equations. (English) Zbl 1349.35276

Neustupa, Jiří

Math. Bohem. 139, No. 4, 685-698 (2014).

Summary: We deal with a suitable weak solution \((\mathbf v,p)\) to the Navier-Stokes equations in a domain \(\Omega \subset \mathbb R^3\). We refine the criterion for the local regularity of this solution at the point \((\mathbf fx_0,t_0)\), which uses the \(L^3\)-norm of \(\mathbf v\) and the \(L^{3/2}\)-norm of \(p\) in a shrinking backward parabolic neighbourhood of \((\mathbf x_0,t_0)\). The refinement consists in the fact that only the values of \(\mathbf v\), respectively \(p\), in the exterior of a space-time paraboloid with vertex at \((\mathbf x_0,t_0)\), respectively in a “small” subset of this exterior, are considered. The consequence is that a singularity cannot appear at the point \((\mathbf x_0,t_0)\) if \(\mathbf v\) and \(p\) are “smooth” outside the paraboloid.

MSC:

35Q30	Navier-Stokes equations
35B65	Smoothness and regularity of solutions to PDEs
35D30	Weak solutions to PDEs
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

Navier-Stokes equation; suitable weak solution; regularity