$$\bex \int_0^T\frac{\sen{\bbu}_{L^{q,\infty}}^p}{\ve+\ln \sex{e+\sen{\bbu}_{L^\infty}}}\rd s<\infty. \eex$$
[Papers]NSE, $u$, Lorentz space [Bjorland-Vasseur, JMFM, 2011]

$$\bex \int_0^T\frac{\sen{\bbu}_{L^{q,\infty}}^p}{\ve+\ln \sex{e+\sen{\bbu}_{L^\infty}}}\rd s<\infty. \eex$$