 Cubic gradient nonlinearity

In three dimensions the map $N(u)=|Du{|}^{2}u$ sends $H^2$ into $L^2$, because $H^2\hookrightarrow W^{1,4}\cap L^{\infty }$. On an $H^2$ ball of radius $R$ it is locally Lipschitz with constant $O(R^2)$.