Local Boundary Regularity for the Navier–Stokes Equations in Non-Endpoint Borderline Lorentz Spaces

Abstract

Local regularity up to the flat part of the boundary is proved for certain classes of distributional solutions that are L_∞L^3,q with q finite. The corresponding result for the interior case was recently proved by Wang and Zhang, see also Phuc’s paper. 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 it to L^3,q with q finite. New scale-invariant bounds, refined pressure decay estimates near the boundary and development of a convenient new ϵ-regularity criterion, are central themes in providing this extension.