## 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].

### MSC:

 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