Geodesic
Local boundary regularity for the Navier--Stokes equations in nonendpoint borderline Lorentz spaces
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 45, Tome 444 (2016), pp. 15-46

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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. 
@article{ZNSL_2016_444_a1,
     author = {T. Barker},
     title = {Local boundary regularity for the {Navier--Stokes} equations in nonendpoint borderline {Lorentz} spaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {15--46},
     publisher = {mathdoc},
     volume = {444},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_444_a1/}
}
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T. Barker. Local boundary regularity for the Navier--Stokes equations in nonendpoint borderline Lorentz spaces. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 45, Tome 444 (2016), pp. 15-46. http://geodesic.mathdoc.fr/item/ZNSL_2016_444_a1/