1. Leray-Hopf $u\in L^\infty(0,T;L^2(\bbR^3))\cap L^2(0,T;H^1(\bbR^3))$. See [Leray, Jean. Sur le mouvement d'un liquide visqueux emplissant l'espace. (French) Acta Math. 63 (1934), no. 1, 193--248].

2. $\om=\n\times u\in L^\infty(0,T;L^1(\bbR^3))$. See for example [Qian, Zhongmin. An estimate for the vorticity of the Navier-Stokes equation. C. R. Math. Acad. Sci. Paris 347 (2009), no. 1-2, 89--92].