The authors give sufficient conditions for boundary regularity of boundary suitable weak solutions of the Navier-Stokes system \[ \begin{aligned} \partial_t v + (v . \nabla)v - \triangle v + \nabla p = 0 \\ \nabla . v = 0 \tag{1} \end{aligned} \] on \(Q_T = \Omega \times (0,T)\) for \(\Omega \subset \mathbb R^3\) and function \(v\) satisfying homogeneous Dirichlet conditions on \(\Gamma \times (0,T)\) for \(C^2\) uniform part \(\Gamma\) of \(\partial \Omega\).
By a boundary suitable weak solution of (1) near \(\Gamma\) the authors understand a pair \((v,p)\) such that \[ v \in L^{2,\infty}\cap W_2^{1,0}\cap W^{2,1}_{\frac{9}{8}, \frac{3}{2}},\quad p \in L^{\frac{3}{2}}\cap W^{1,0}_{\frac{9}{8}, \frac{3}{2}}, \] functions \(v,p\) satisfy (1) a.e. on \(Q_T\), local energy inequality holds near \(\Gamma\) in the following sense: \[ \begin{split}\int_{\Omega} \xi(y,t)|v(y,t)|^2 dy + 2\int_0^t \int_{\Omega}\xi |\nabla v|^2 dy d\tau \\ \leq \int_0^t\int_{\Omega}\{|v|^2(\partial_t \xi + \triangle \xi)+ v .\nabla \xi(|v|^2 + 2p)\} dy d\tau \end{split} \] for a.e. \(t \in (0,T)\) and all nonnegative functions \(\xi \in C_0^{\infty}(\mathbb R^3\times(0,T))\) vanishing near \((\partial \Omega \setminus \Gamma)\times(0,T)\).
The main result of the paper reads as follows: Let \(\Gamma\) be \(C^2\) uniform, \(x_0 \in \Gamma, z_0 = (x_0, t_0)\) and let \((v,p)\) be a boundary suitable weak solution of (1) near \(\omega(z_0, r)\cap \Gamma\) where \[ \omega(z_0,r)= \left(\Omega \cap B(x_0,r)\right)\times(t_0-r^2, t_0). \] Assume additionally that \(v \in L^{3,\infty}(\omega(z_0,r))\). Then \(v\) is Hölder continuous on \(\overline{\omega(z_0,r/2)}\).
The idea of the proof follows the procedure used by L. Escauriaza, G. A. Serëgin and V. Shverak [Russ. Math. Surv. 58, No. 2, 211–250 (2003); translation from Usp. Mat. Nauk 58, No. 2, 3–44 (2003; Zbl 1064.35134); in: Nonlinear problems in mathematical physics and related topics II. In honour of Professor O. A. Ladyzhenskaya. New York, NY: Kluwer Academic Publishers. Int. Math. Ser., N.Y. 2, 353–366 (2002; Zbl 1024.76011)]. Assume \(z_0\) be a point of singularity. By suitable scaling and blow up the authors obtain “global” and “ancient” solution, i.e. boundary suitable weak solution \(u\) on \(\mathbb R^3_+ \times (\infty,0)\). Moreover, \(u\) vanishes at time \(t = 0\). Then curl \(u\) satisfies backward heat equation and by [Zbl 1064.35134] vanishes identically which is a contradiction.