Let

 - $d\in\left\{2,3\right\}$ with 
 - $\Lambda\subseteq\mathbb R^d$ be bounded and open with $\partial\Lambda\in C^1$

In [Lemma 6.1](https://books.google.de/books?id=W9shWJXh-SEC&pg=PA44&lpg=PA44&dq=%22since+the+first+derivatives+of%22+%22are+continuous+on%22&source=bl&ots=wgz76uXDbe&sig=Y8nT5AH6RbktZk-wXdRQTupAXCE&hl=de&sa=X&ved=0ahUKEwiNpbfcg-_QAhXHFCwKHUEdAcUQ6AEIHTAA) of *Navier-Stokes Equations and Nonlinear Functional Analysis* by *Roger Temam*, the author is stating that if $u,v\in H^2(\Lambda,\mathbb R^d)$, then $$\frac\partial{\partial x_i}(u\cdot\nabla)v=\left(\frac{\partial u}{\partial x_i}\cdot\nabla\right)v+(u\cdot\nabla)\frac{\partial v}{\partial x_i}\tag 1$$ would belong to $L^2(\Lambda,\mathbb R^d)$, because $u,v$ belong to $L^6(\Lambda,\mathbb R^d)$ (which is clear by the Sobolev inequalities) **and $u,v$ are continuous on $\overline\Lambda$**.

I don't understand why $u,v$ need to be continuous on $\overline\Lambda$ (I know that they are Hölder continuous in the case $d=1$) and I don't understand how he's using this to obtain the claim.