Criteria for the regularity of the solutions to the Navier–Stokes equations based on the velocity gradient

We study the regularity of solutions to the Navier–Stokes equations in the whole three-dimensional space under the assumption that some additional conditions are imposed on one or more entries of the velocity gradient. Many such results with conditions using the Lebesgue spaces can be found in the literature, starting with the classical Beirao da Veiga’s result. The main goal of the present paper is to generalize the known results by replacing the standard Lebesgue spaces by wider Besov spaces in space variables. In most such cases the technique of the proof leads to the deterioration of the Prodi–Serrin scale, so the results with the Besov spaces are not still quite satisfactory. Nevertheless, our technique leads to extensions and improvements of some results from the literature.