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We consider the vanishing viscosity limit of the Navier--Stokes equations in a half-plane, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the product of the components of the Navier--Stokes solutions are equicontinuous at $x_2=0$. A sufficient condition for this to hold is that the tangential Navier--Stokes velocity remains uniformly bounded and has a uniformly integrable tangential gradient near the boundary.

Remarks on the Inviscid Limit for the Navier--Stokes Equations for Uniformly Bounded Velocity Fields