Navier-Stokes equations: almost \(L_{3,\infty}\)-case. (English) Zbl 1128.35085

Seregin, G.

J. Math. Fluid Mech. 9, No. 1, 34-43 (2007).

The regularity of solutions to the Navier-Stokes equations is studied. Let the pair \((v,p)\) be the solution to the Navier-Stokes equations in \[ \frac{\partial v}{\partial t}+(v\cdot\nabla)v-\Delta v+\nabla p=0,\quad \text{div}\,v=0, \] in \(Q_T=\Omega\times(0,T)\), \(\Omega\subset\mathbb R^3\), \(T>0\), \(v\in L_{2,\infty}(Q_T)\cap W^{1,\,0}_2(Q_T)\), \(p\in L_{3/2}(Q_T)\).
The author proves that if \[ \lim_{t\to T}\inf\frac{1}{T-t}\int_t^T\int_\Omega | v(x,s)| ^3\,dx\,ds<\infty \] then the solution is regular on the set \(\{x\in\Omega, t=T\}\) in the sense of suitable weak solutions [see G. A. Seregin, Algebra Anal. 14, No. 1, 194–237 (2002; Zbl 1039.35080)].

Reviewer: Il’ya Sh. Mogilevskij (Tver’)

MSC:

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

Keywords:

Navier-Stokes equations; suitable weak solutions; regularity

Citations:

Zbl 1039.35080