Local boundary regularity for the Navier–Stokes equations in nonendpoint borderline Lorentz spaces

We prove local regularity up to the flat part of the boundary, for certain classes of distributional solutions that are $L_\infty L^{3,q}$ with $q$ finite. The corresponding result, for the interior case, was proven recently by Wang and Zhang, see also work by Phuc. For local regularity, up to the flat part of the boundary, $q=3$ was established by G. A. Seregin. Our result can be viewed as an extension of this to $L^{3,q}$ with $q$ finite. New scale-invariant bounds, refined pressure decay estimates near the boundary and development of a convenient new $\epsilon$-regularity criterion are central themes in providing this extension.