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2009-13

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

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