On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. (English) Zbl 0954.35129

Ladyzhenskaya, O. A.; Seregin, G. A.

J. Math. Fluid Mech. 1, No. 4, 356-387 (1999).

The authors consider certain weak solutions of the Navier-Stokes equations in a bounded domain \(\Omega\subset \mathbb{R}^3\), i.e. \[ \begin{alignedat}{2} v_t-\Delta v+ (v\cdot\nabla)v+\nabla p & =f &&\quad\text{in }\Omega\times (0,T),\\ \text{div }v &= 0 &&\quad\text{in }\Omega\times (0,T),\\ v &= 0 &&\quad\text{on }\partial\Omega\times (0,T),\\ v(0) &= a &&\quad\text{in }\Omega.\end{alignedat} \] It is assumed that the velocity \(v\) belongs to \(L^\infty(0,T; L^2(\Omega))\cap L^2(0, T; W^1_2(\Omega))\), \(p\in L^{{3\over 2}}(\Omega\times (0,T))\) and that \((v,p)\) satisfies a localized energy inequality. The main result of the paper is a partial regularity theorem, which says that if the quantity \[ \limsup_{R\to 0} {1\over R} \int^{t_0}_{t_0- R^2} \int_{B_R(x_0)}|\nabla v|^2 \] is sufficiently small at a point \((x_0,t_0)\in \Omega\times (0,T)\), then \((x,t)\mapsto v(x,t)\) is Hölder continuous near \((x_0,t_0)\). Furthermore, the complement of the points having the above property has parabolic Hausdorff dimension \(\leq 1\). The proof is based on a blow-up argument combined with a special invariant structure of the equations.

Reviewer: Klaus Deckelnick (Brighton)

MSC:

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

Keywords:

Navier-Stokes equations; partial regularity; Hausdorff dimension