09-13
On the boundary regularity of suitable weak solutions to the Navier-Stokes equations
by Wolf, J.
Preprint series: 09-13, Preprints
- MSC:
- 35Q30 Stokes and Navier-Stokes equations, See also {76D05, 76D07, 76N10}
- 35D10 Regularity of generalized solutions
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 $p$.
Keywords: Navier-Stokes equations, partial regularity
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