How can I prove that
$$\int \int_{ \mathbb{R}^+\times U} \left( \frac{ \partial m_n}{\partial t}-m_n\times \frac{ \partial m_n}{\partial t}\right) \Phi \to \int \int_{\mathbb{R}^+\times S^1} \left( \frac{ \partial m}{\partial t}-m\times \frac{ \partial m}{\partial t}\right) \Phi, \forall \Phi \in \mathcal{D}(\mathbb{R}^+; H^1(U))$$
Where $U= (S^1(0,1)\times B^2(0,1))$, $m $ a vector in $\mathbb{R}^3$, $\times $ is the vector product and $$ m_n \to m \text{ weak*} \text{ in } L^{\infty}(0,T,H^1(U)) \text{ and fort in } L^{\infty}(0,T,L^p(U)), 2<p\leq 6,$$ $$ \frac{ \partial m_n}{\partial t} \to \frac{ \partial m}{\partial t} \text{ weak in } L^2(0,T, L^2(U) ) $$
You can prove that $ m_n \times \frac {\partial m_n}{\partial t} $ converges weakly to $m \times \frac {\partial m}{\partial t}$ in $ L^2(0,T;L^q(U)) $ with $\frac12 + \frac1p+\frac1q=1$. Use something like $a_nb_n -ab=(a_n-a)b_n + a(b_n-b)$ and Hoelder inequality.