# Otto-von-Guericke-Universität Magdeburg

### 09-13

#### OntheboundaryregularityofsuitableweaksolutionstotheNavier-Stokesequations

by Wolf, J.

Preprint series: 09-13, Preprints

MSC:
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 $\abla \bf{u}$ and the $L^{3/2}$-norm of the pressure $p$.