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.
Best Answer
As you noted, the authors use single bars $|\cdot |$ for Hilbert space norm, without any subscript. And they use $|\cdot |$ for pointwise vector norm as well, as in (2.11). This made me look for the downvote button, but the journal's site does not have one. Anyway, the first step after the triangle inequality is Cauchy-Schwarz: $$\|u\cdot \nabla C\|_{L^2(\Omega)}=\left(\int_\Omega |u\cdot \nabla C|^2\right)^{1/2} \le \left(\int_\Omega |u|^4\right)^{1/4}\left(\int_\Omega |\nabla C|^4\right)^{1/4}$$ The second step is the estimate $$\|u\|_{L^4}\le M\|u\|_{L^2}^{1/2}\|\nabla u\|_{L^2}^{1/2}\tag{1}$$ (and the same for $\nabla C$). At first glance, (1) looks like the Sobolev embedding combined with trivial interpolation between $L^2$ and $L^\infty$ but it's not because $H^1$ does not embed into $L^\infty$ (in nice planar domains it embeds into $L^p$ for $p<\infty$). However, (1) is still true: this is a special case of Gagliardo-Nirenberg interpolation inequality, stated, for example, on page 314 of the book Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis. Brezis does not give a proof and refers the reader to Friedman's book which I don't have... but I just noticed it was republished by Dover! Extra special +1 to your question for this.