Interior regularity criteria in weak spaces for the Navier-Stokes equations (Q706160)

The authors examine the interior regularity of weak Leray-Hopf solutions of unsteady incompressible Navier-Stokes equations in a three-dimensional domain \(\Omega\). The local boundedness of a weak solution \(u\) in open parabolic balls is proved under the assumption that \(\| u\|_{L^s_w(0,T; L^r_w(\Omega))}\) is sufficiently small for some \(r\), \(s\) with \({2\over s}+ {3\over r}= 1\) and \(3\leq r<\infty\). The novelty of the paper is the extension of numerous previous results to the critical case \(r=3\). A brief discussion of the second critical case \(r=\infty\) is presented in the final section.