New sufficient conditions for regularity of solutions to the Navier-Stokes equations. (English) Zbl 1039.35077

Summary: We show that a weak solution \(u\) to the 3D Navier-Stokes equations belongs to the space \(L^2(0,T;L^\infty({\mathbb R}^3))\) if one of the following three conditions holds:
(i) \[ \int^T_0(\sup_{x_0\in{\mathbb R}^3}\int_{{\mathbb R}^3}(|x-x_0|^\alpha u)^qdx)^{s/q}dt<+\infty \] for some \(\alpha\in\mathbb R\) and \(q>1\), with \(1/s+3/2q=1/2-\alpha/2,\;\alpha+1\geq0\) and \(-1<\alpha+3/q<1\);
(ii) There exists \(r_0>0\) such that \[ \epsilon=\sup_{t\in[0,T]}\sup_{x_0\in {\mathbb R}^3}\int_{B_{r_0}(x_0)}|x-x_0|^{-1}|u|^2dx \] is sufficiently small;
(iii) There exists \(r_0>0\) such that \[ \int^T_0\sup_{x_0\in{\mathbb R}^3}\int_{B_{r_0}(x_0)}|x-x_0|^{-1}|\nabla u|^2dxdt<+\infty. \]


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