Let $ \Omega$ be open and bounded. Consider that $u$ is smooth solution of \begin{equation} \frac{\partial u}{\partial t}=(u\cdot \nabla)u~~~in~\Omega\times[0,T]\\ u=0~on~~~\partial \Omega\times[0,T]\\ u(\cdot,0)=0~in~\Omega. \end{equation}
Notice that $u$ is not divergence-free. After testing with $u$: \begin{equation} \frac{d}{2dt}||u||_2^2=\int_\Omega (u\cdot \nabla)u\cdot u~dx \end{equation} I have a problem due to convective term. I want to get an estimate on which one can apply Gronwall lemma.