Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary. (English) Zbl 0997.35044

Seregin, G. A.

J. Math. Fluid Mech. 4, No. 1, 1-29 (2002).

The Navier-Stokes equations \[ \frac{\partial v}{\partial t}-\Delta v+ v\cdot\nabla v+\nabla p=f, \quad \nabla\cdot v=0 \] are considered in the cylindrical domain \(Q_T=\Omega\times(0, T),\;\Omega\subset \mathbb R^3\). Suitable weak solutions to the Navier-Stokes equations were investigated in L. Caffarelli, R.-V. Kohn and L. Nirenberg [Commun. Pure Appl. Math. 35, 771–831 (1982; Zbl 0509.35067)]. Shortly speaking, the suitable weak solution is a weak solution satisfying the local energy inequality.
It is proved in the present paper that the suitable weak solution \(v\) under some reasonable restrictions satisfies the Hölder condition in a neighborhood of \(z_0=(x_0, t_0)\in Q_T,\;x_0\in\partial\Omega\), at least if there exists a neighborhood \(U\) of the point \(x_0\in\partial\Omega\) such that \(\Gamma=\partial\Omega\cap U\) lies in a hyperplane and \(v=0\) on \(\Gamma\times [0, T]\). The proof is based on a new decomposition of the pressure \(p\) which is very perspective and productive.

Reviewer: Il’ya Sh. Mogilevskii (Tver)

MSC:

35Q30	Navier-Stokes equations
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
35D30	Weak solutions to PDEs

Keywords:

Navier-Stokes equations; initial-boundary value problem; suitable weak solution; Hölder continuity

Citations:

Zbl 0509.35067