by Wolf, J..

**Series:** 2009-13, Preprints

- MSC:
- 35Q30 ~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 $\mathbb{R}^3_+$. Our main result is a direct proof of the partial regularity up to the flat boundary based upon a new decay estimate, which implies the regularity in the cylinder $Q^+_\rho(x_0,t_0)$ provided

\[

\limsup_{R \to 0} \frac{1}{R} \int_{Q^+_\rho(x_0,t_0)}

|\rm rot\,{\bf u}|^2 dx dt \le \varepsilon_0

\]

with $\varepsilon $ sufficiently small. In addition, we present a new condition for the local regularity beyond Serrin's class which involves the $L^2$-norm of $\nabla \bf{u}$ and the $L^{3/2}$-norm of the pressure $p$.

**Keywords:**

Navier-Stokes equations, partial regularity