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

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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 on a new decay estimate, which implies the regularity in the cylinder \({Q_\rho ^+(x_0, t_0)}\) provided

$$\limsup_{R\to 0}\frac {1} {R}\int\limits_{Q_R^+(x_0, t_0)} |{\rm rot}\,\mathbf u|^2 dxdt \,\leq\, \varepsilon _0$$

with ε 0 sufficiently small. In addition, we get a new condition for the local regularity beyond Serrin’s class which involves the L 2-norm of ∇u and the L 3/2-norm of the pressure.

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  1. Otto-von-Guericke University Magdeburg, Magdeburg, Germany

    Jörg Wolf

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  1. Jörg Wolf
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Wolf, J. On the boundary regularity of suitable weak solutions to the Navier-Stokes equations. Ann. Univ. Ferrara 56, 97–139 (2010). https://doi.org/10.1007/s11565-010-0091-3

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