Weak convergence and compacity

I have:

$$\parallel \nabla m_n \parallel_{L^{\infty}(\mathbb{R}^+, L^2(\Omega))}\leq C$$

$$\parallel \frac{\partial m_n}{\partial z} \parallel_{L^{\infty}(\mathbb{R}^+, L^2(\Omega))}\leq C n$$

$$\parallel \frac{\partial m_n}{\partial t} \parallel_{L^2(\mathbb{R}^+\times\Omega)}\leq C$$

$$\parallel m_{3,n} \parallel_{L^{\infty}(\mathbb{R}^+, L^2(\Omega))}\leq C \sqrt{n}$$

So we have for a subsequence,

$$m_n \to m$$ weakly * in $$L^{\infty}(\mathbb{R}^+, H^1(\Omega))$$

$$\partial_z m_n \to 0$$ in $$L^{\infty}(\mathbb{R}^+, L^2(\Omega))$$

$$m_{3,n} \to 0$$ in $$L^{\infty}(\mathbb{R}^+, L^2(\Omega))$$

$$\partial_t m_n \to \partial_t m$$ in $$L^2([0,T],\times\Omega)$$ for any $$T>0$$

my question why it follows that:

$$m_n \to m$$ in $$L^{2}([0,T]\times \Omega)$$ for any $$T>0$$ and in $$L^{p}([0,T]\times \Omega)$$ for any $$p \geq 1$$. and $$m_n(0) \to m(0)$$ weakly in $$(L^2(\Omega))^3$$ and $$\partial_t m_n,3 \to$$ weakly in $$L^2([0,T]\times \Omega)$$ for any $$T>0$$

$$\nabla m_{n,3} \to 0$$ weakly in $$L^2([0,T] \times \Omega)$$ for any $$T>0$$.