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 $L^6(\Lambda,\mathbb R^d)$ and $u,v$ are continuous on $\overline\Lambda$ (which is clear by the Sobolev inequalities as they can be found, for example, in the [book of Evans](https://books.google.de/books?id=Xnu0o_EJrCQC&pg=PA286&lpg=PA286&dq=%22evans%22+%22General+Sobolev+inequalities%22+%22be+a+bounded+open%22&source=bl&ots=dkidsCTqpS&sig=Nn7_q90n09Sw8TQS8vTQhJ4IfDI&hl=de&sa=X&ved=0ahUKEwi9yvGWp-_QAhUBCSwKHQSHDm8Q6AEIIzAB#v=onepage&q=%22evans%22%20%22General%20Sobolev%20inequalities%22%20%22be%20a%20bounded%20open%22&f=false)).

Can the same statement be proved, if $\Lambda$ is just bounded and open and $u,v\in H_0^2(\Lambda,\mathbb R^d)$?