$u$}(\cdot,s)\!\;\!\cdot \!\;\!\nabla
\mbox{\boldmath $u$}\;\!(\cdot,s)\;\!]\:\!(k)
\:|_{\mbox{}_{2}}
$$\
para todo $\;\! k \in \mathbb{R}^{3}\!$, de modo que\
\
$$\tag{2.9}
\|\:
\hat{\mbox{\boldmath $Q$}}(\cdot,s)
\,\|_{\mbox{}_{\scriptstyle \infty}}
\;\!\leq\;
\|\;
\mathbb{F}\;\![\, \mbox{\boldmath $u$}
\!\;\!\cdot \!\;\!\nabla
\mbox{\boldmath $u$}\,]\:\!(\cdot,s)
\:\|_{\mbox{}_{\scriptstyle \infty}}.$$\
Por outro lado, tem-se, para $ 1 \leq i \leq 3 $,\
\
$$\notag
\begin{split}
|\; \mathbb{F}\;\![\, \mbox{\boldmath $u$}(\cdot,s)
\!\;\!\cdot \!\;\!\nabla
\;\!u_{{\scriptstyle i}}(\cdot,s)\,]\:\!(k) \:|
\;\,&\leq\;
\sum_{j\,=\,1}^{3} \;\!
|\; \mathbb{F}\;\![\, u_{{\scriptstyle j}}(\cdot,s) \;\!
D_{{\scriptstyle \!\;\!j}} \:\! u_{{\scriptstyle i}}(\cdot,s)
\,]\:\!
(k) \:| \\
&\leq\;
(\:\!2\:\!\pi)^{-\,3/2} \;\!
\sum_{j\,=\,1}^{3} \;\!
\|\, u_{{\scriptstyle j}}(\cdot,s) \;\!
D_{{\scriptstyle \!\;\!j}} \:\! u_{{\scriptstyle i}}(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{3})}} \\
&\leq\;
(\:\!2\:\!\pi)^{-\,3/2} \,
\|\, \mbox{\boldmath $u$}(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
\;\!
\|\, \nabla \!\;\! u_{{\scriptstyle i}}(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
\!\;\!,
\end{split}
$$\
usando Cauchy-Schwarz. $\!$($\;\!$Aqui, como sempre, $ D_{\!\;\!j} \!\;\!=\,\! \partial/\partial x_{\!\;\!j} \!\;\!$.) Isso fornece\
\
$$\tag{2.10}
\|\; \mathbb{F}\;\![\, \mbox{\boldmath $u$} \!\;\!\cdot \!\;\!\nabla
\;\!\mbox{\boldmath $u$}\,]\:\!(\cdot,s) \:\|_{\mbox{}_{\scriptstyle \infty}}
\:\leq\;
(\:\!2\:\!\pi)^{-\,3/2} \,
\|\, \mbox{\boldmath $u$}(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
\;\!
\|\, D \mbox{\boldmath $u$}(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
\!\:\!.$$\
De (2.7), (2.9) and (2.10), obtém-se (2.5), concluindo a prova da Proposição 2.1. ]{} $\Box$\
\
Note-se que, repetindo o argumento acima para as soluções das equações regularizadas (2.1), obtém-se de modo análogo que\
\
$$\tag{2.11}
\|\: e^{\;\!\nu \:\!
\mbox{\scriptsize $\Delta$} \mbox{\scriptsize $(t - 