Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes equations. (English) Zbl 1027.35094

Neustupa, Jiří(ed.) et al., Mathematical fluid mechanics. Recent results and open questions. Basel: Birkhäuser. Adv. Math. Fluid Mech. 237-268 (2001).
The following Navier-Stokes initial-boundary value problem for a viscous incompressible fluid with the homogeneous Dirichlet-type boundary condition is considered: \[ {\partial v\over\partial t}+(v \cdot \nabla) v=f-\nabla p+\nu \Delta v,\quad \text{div} v=0 \text{ in }Q_T, \]
\[ v=0 \text{ on }\partial \Omega\times] 0,T[,\quad v|_{t=0}= v_0, \] where \(\Omega\) is a domain in \(\mathbb{R}^3\), \(T\) is a positive number, \(Q_T=\Omega \times]0,T[\), \(v=(v_1,v_2,v_3)\) and \(p\) denote the unknown velocity and pressure, resp., \(f\) is an external force and \(\nu>0\) is the viscocity coefficient. The existence of a unique regular solution for a general three-dimensional flow is so far known only locally in time or under the assumption that \(v_0\) and \(f\) are in appropriate norms “small enough”.
The existence of a regular solution of the problem on the time interval \(]0,T[\) of an arbitrary length \(T\), in the case of \(v_0\) and \(f\) of an arbitrary size, remains an open problem.
In this paper, sufficient conditions for regularity of a suitable weak solution \((v, p)\) in a sub-domain \(D\) of the cylinder \(Q_T\) are formulated. The conditions are anisotropic in the sense that the assumptions about \(v_1\) and \(v_2\) differ from the assumptions about \(v_3\). The question what types of deformations of infinitely small volumes of the fluid support regularity and what types contribute to a blow up is studied too.
For the entire collection see [Zbl 0971.00052].


35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35D10 Regularity of generalized solutions of PDE (MSC2000)
76D05 Navier-Stokes equations for incompressible viscous fluids