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Boundary regularity criteria for the 6D steady Navier-Stokes and MHD equations. (English) Zbl 1378.35221
Summary: It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes and MHD equations are Hölder continuous near boundary provided that either \(r^{-3}\int_{B^+_r}|u(x)|^3dx\) or \(r^{-2}\int_{B^+_r}|\nabla u(x)|^2dx\) is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points near the boundary is zero. This generalizes recent interior regularity results by H. Dong and R. M. Strain [Indiana Univ. Math. J. 61, No. 6, 2211–2229 (2012; Zbl 1286.35193)].
MSC:
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35D30 Weak solutions to PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
76D05 Navier-Stokes equations for incompressible viscous fluids
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