In his paper 'Nonstationary flows of viscous and ideal fluids in $\Bbb{R}^3$' Kato mentioned the following: a limit that exists strongly in $H^{m-1}$ and weakly in $H^{m}$ what is explicitly the meaning of that convergence? is it simply the following: $$\lim\limits_{\nu\to 0}\|u_\nu-u_0\|_{H^{m-1}}=0~~\mbox{and} ~\lim\limits_{\nu\to0}\langle u_\nu-u,\phi\rangle_{H^m}=0,$$ where $\phi$ is a test function (well localized and a regular enough function)
Kato, Tosio, Nonstationary flows of viscous and ideal fluids in $R^3$, J. Funct. Anal. 9, 296-305 (1972). ZBL0229.76018.