On the boundary regularity of suitable weak solutions to the Navier-Stokes equations

Abstract

We consider suitable weak solutions to an incompressible viscous Newtonian fluid governed by the Navier-Stokes equations in the half space R 3 + . Our main result is a direct proof of the partial regularity up to the flat boundary based on a new decay estimate, which implies the regularity in the cylinder Qwith ε 0 sufficiently small. In addition, we get a new condition for the local regularity beyond Serrin's class which involves the L 2 -norm of ∇u and the L 3/2 -norm of the pressure.

Boundary partial regularity for the high dimensional Navier–Stokes equations

“…In the 5D stationary case, Kang [10] proved the partial regularity up to the boundary, which extended the result by Struwe [28]. See also recent Wolf [32], Mikhailov [17], and the references therein. Motivated by these paper, the objective of the current paper is to extend the aforementioned interior estimates in [3,4] to the boundary case.…”

Boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations

“…Other types of conditions in terms of scaled invariant norms near boundary are also found in [35] (compare to [24], [26], [13], [31], [8], [27], [38] for the NSE). We also refer to [11], [15] and [34] in the interior case for MHD equations (compare to [23], [1], [32], [33], [19], [17], [9] for the NSE).…”

On Regularity of a Weak Solution to the Navier–Stokes Equations with the Generalized Navier Slip Boundary Conditions

“…(This concerns, e.g., the criteria from [2,12].) As to criteria, valid up to the boundary, we can cite, for example, the papers [13] (where the socalled suitable weak solution is shown to be bounded locally near the boundary if it satisfies Serrin's conditions near the boundary and the trace of the pressure is bounded on the boundary), [14] (where an analogy of the well-known Caffarelli-Kohn-Nirenberg criterion for the regularity of a suitable weak solution at the point (x 0 , 0 ) ∈ Ω × (0, ), e.g., [15], is also proven for points on a flat part of the boundary), and [16,17] (for some generalizations of the criterion from [14], however, also valid only on a flat part of the boundary). A generalization of the criterion from [14] for points (x 0 , 0 ) on a "smooth" curved part of the boundary can be found in paper [18].…”