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We study the regularity criteria for weak solutions to the $3D$incompressible Navier--Stokes equations in terms of the geometry of vortexstructures, taking into account the boundary effects. A boundary regularitytheorem is proved on regular domains with a class of oblique derivativeboundary conditions, providing that the vorticity of the fluid is coherentlyaligned. In particular, we establish the boundary regularity on round balls,half-spaces and right circular cylindrical ducts, subject to the classicalNavier and kinematic boundary conditions.

Geometric Regularity Criteria for Incompressible Navier--Stokes Equations with Navier Boundary Conditions