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On local regularity conditions for the Navier–Stokes equations

We address the interior regularity of suitable weak solutions of the 3D incompressible Navier–Stokes equations. We prove a partial regularity criterion based on only one component of the velocity. Furthermore, the uniform estimate obtained in this criterion allows us to prove a generalization of the Leray regularity condition. Finally, we extend the regularity due to Escauriaza–Serëgin–Šverák. Namely, we prove that if is a suitable weak solution and one of the velocity components belongs to , while the remaining two are in the larger Morrey space , and if the pressure satisfies , then the solution is regular.