Suppose $\Omega \subset \mathbb{R}^2$ is a bounded open set with $C^1$ boundary. Suppose $u \in W^{1,3}(\Omega)$. Then prove that $u^2 \in W^{1,3}(\Omega)$ and we have the following estimate,
$$ ||u^2||_{W^{1,3}(\Omega)} \leq C||u||_{W^{1,3}(\Omega)}^2 $$ for some $C > 0$.
Proving that $u^2 \in W^{1,3}(\Omega)$ was a simple application of the chain rule for weak derivatives. For instance, see here: Chain rule in the Sobolev space $W^{1,p}$.
I'm having difficulty with the estimate, however. Since $D(u^2) = 2uDu$, this amounts to showing that,
$$ ||u^2||^3_{L^3(\Omega)} + ||2uDu||^3_{L^3(\Omega)} \leq C\left(||u||_{L^3(\Omega)}^3 + ||Du||_{L^3(\Omega)}^3\right)^2 $$
I don't think we can use any embedding theorems here because we are in the case where $p = 3 > 2 = n$. Morrey's inequality would apply but that involves Holder norms. Are there any $L^p$ norm manipulations I can use like a variant of Holder's inequality?
Morrey's inequality is in fact exactly what you need. If $u \in W^{1,p}(\Omega)$, $\Omega \subset \mathbb R^n$, $p > n$, and $\lambda = 1 - \frac np$ then $u \in C^{0,\lambda}(\Omega)$ and $\|u\|_{C^{0,\lambda}(\Omega)} \le C \|u\|_{W^{1,p}(\Omega)}$. This validity of this inequality is the only point where the boundary condition on $\Omega$ is needed.
Functions in $C^{0,\lambda}$ are bounded. In fact $\sup_\Omega |u| \le \|u\|_{C^{0,\lambda}(\Omega)}$.
That's about it. If $u \in W^{1,3}(\Omega)$ and $n=2$ then $$\|u^2\|_{L^3(\Omega)} \le |\Omega|^{1/3} \sup_\Omega |u|^2 \le |\Omega|^{1/3} \|u\|^2_{C^{0,1/3}(\Omega)} \le C |\Omega|^{1/3}\|u\|^2_{W^{1,3}(\Omega)} $$ and $$\|uDu\|_{L^3(\Omega)} \le \sup_\Omega |u| \|Du\|_{L^3(\Omega)} \le \|u\|_{C^{0,1/3}(\Omega)}\|u\|_{W^{1,3}(\Omega)} \le C \|u\|^2_{W^{1,3}(\Omega)}.$$