On a regularity criterion for the solutions to the 3D Navier-Stokes equations. (English) Zbl 1034.35087

The author considers initial-boundary problem for Navier-Stokes equations in the whole three-dimensional space or in a three-dimensional domain \(\Omega\) uniformly of class \(C^2\). If the initial data belongs to the closure of compactly-supported divergence-free vector fields in the norm \(H^1_0(\Omega)\) and \(\nabla u\in L^\alpha (0,T;L^\beta (\Omega))\) for \(\frac{2} {\alpha} +\frac{3} {\beta}=2\), \(1\leq\alpha\leq 2\), \(3\leq\beta\leq\infty\), then the main results states that the solution \(u\) is strong in \([0;T]\).


35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids