I need a small help in understanding the following that how "Nirenberg -Gagliardo-Sobolev inequalities" were used. This is a part of the paper.
Denote $ H^1=W^{1, 2}(\Omega)\\ V_1=\{ f\in H^2 (\Omega) : \frac{\partial f}{\partial n}|_{\partial \Omega} =0\}\\ H_2=\{ f\in L^2(\Omega)^2: \nabla . f =0,~ f.n=0 \text{ on } \partial \Omega\}\\ V_2 = \{ f\in H^1_0(\Omega)^2 : \nabla . f =0\} $
Let $C \in L^2(0, \tau ; V_1) \cap L^\infty (0, \tau; H^1)$ and $u \in L^2(0, \tau ; V_2) \cap L^\infty (0, \tau; H_1)$ are bounded.
By $|\cdot|$ we denote the norm on both in $L^2(\Omega)$ and $l^2(\Omega)^2$ where $\Omega$ a bounded regular open set in $\mathbb{R^2}$, with boundary $\partial \Omega$.
Now given equation is $ C'_t = d \Delta C - u . \nabla C$ we have
$ |\frac{\partial C}{\partial t}| \leq | d \Delta C| + |u . \nabla C|\qquad \text{ triangle inequality} $ $ \leq d\|\Delta C | ~+~\|u\|_{L^4(\Omega)^2} \|\nabla C\|_{L^4(\Omega)^2}\qquad\text{How?} $ $ \leq d\|\Delta C | + M |u|^{1/2} \|u\|^{1/2}_{H^1(\Omega)^2} \|C\|^{1/2}_{H^1(\Omega)} \|C\|^{1/2}_{H^2(\Omega)}\qquad\text{How is this for some $M>0$?} $
I could not proceed how they have use Nirenberg- Gagliardo-Sobolev inequalities or some other inequalities. I badly stuck at this point.