Boundary regularity criteria for the 6D steady Navier–Stokes and MHD equations

Abstract

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 −3is 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 Dong-Strain [5].

Classification of solutions of the 2D steady Navier-Stokes equations with separated variables in cone-like domains

“…Second, our examples show that the singularity of the solution does not depend on the regularity of the boundary (for example, the case of R 2 + when α = 0, β = π). Also, for boundary regularity criteria of steady Navier-Stokes equations (for example, see [10] for the boundary Hölder regularity of 6D steady Navier-Stokes equations), it's impossible to prove the uniform boundary C γ regularity with γ > γ 0 > 0 independent of u due to the examples in (iv) for nontrivial boundary data.…”

Classification of solutions of the 2D steady Navier-Stokes equations with separated variables in cone-like domains

“…The examples in type (iv) show that: for any 0 < γ < 1 some solutions of steady Navier-Stokes equations have the boundary C γ regularity similar to their nontrivial boundary data, while one can prove a uniform boundary C γ0 with γ 0 > 0 regularity result for 6D steady Navier-Stokes equations with zero-Dirichlet boundary data (for example, see [19]). Therefore the boundary data plays an important role in the boundary regularity theory for steady Navier-Stokes equations.…”