s)$}} \;\!
\mbox{\boldmath $Q$}_{\mbox{}_{\scriptstyle \!\delta}}\!\;\!(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
\leq\,
K \,
\nu^{\mbox{}^{\scriptstyle \!-\, \frac{3}{4}}}
(t - s)^{\mbox{}^{\scriptstyle \!\!-\, \frac{3}{4}}}
\|\, \mbox{\boldmath $u$}_{\mbox{}_{\scriptstyle \!\:\!\delta}}
\!\;\!(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
\;\!
\|\, D \mbox{\boldmath $u$}_{\mbox{}_{\scriptstyle \!\:\!\delta}}
\!\;\!(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}$$\
para cada $\;\!t > s > 0 $, sendo $ K \!\;\!=\:\! (\:\!8 \:\!\pi)^{-\,3/4} \!$, como antes, e\
\
$$\tag{2.12}
\mbox{\boldmath $Q$}_{\mbox{}_{\scriptstyle \!\delta}}
\!\:\!(\cdot,s)
\;=\; -\:
\bar{\mbox{\boldmath $u$}}_{\mbox{}_{\scriptstyle \!\:\!\delta}}\!\:\!(\cdot,s)
\!\;\!\cdot \!\;\!\nabla
\mbox{\boldmath $u$}_{\mbox{}_{\scriptstyle \!\:\!\delta}}\!\:\!(\cdot,s)
\,-\;\!
\nabla p_{\mbox{}_{\scriptstyle \!\delta}}\!\;\!(\cdot,s).$$\
A estimativa (2.11) é muito útil, visto que as soluções regularizadas $ {\displaystyle
\:\!
\mbox{\boldmath $u$}_{\mbox{}_{\scriptstyle \!\:\!\delta}}\!\:\!(\cdot,t)
\:\!
} $ definidas em (2.1) satisfazem a desigualdade de energia\
\
$$\tag{2.13}
\|\,\mbox{\boldmath $u$}_{\mbox{}_{\scriptstyle \!\:\!\delta}}
\!\:\!(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}^{\:\!2}
+\;
2 \, \nu \!\!\;\!
\int_{0}^{\;\!\mbox{\mbox{\footnotesize $t$}}} \!\!\;\!
\|\, D \mbox{\boldmath $u$}_{\mbox{}_{\scriptstyle \!\:\!\delta}}
\!\:\!(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}^{\:\!2}
\;\!ds
\;\;\!\leq\:
\|\, \mbox{\boldmath $u$}_{0} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}^{\:\!2}
$$\
para todo $\;\! t > 0 $ (e todo $ \delta > 0 $), de modo que $ {\displaystyle
\;\!
\|\,\mbox{\boldmath $u$}_{\mbox{}_{\scriptstyle \!\:\!\delta}}
\!\:\!(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
\!\;\!
} $, $
\int_{0}^{\;\!\mbox{\mbox{\footnotesize $t$}}} \!
\|\, D \mbox{\boldmath $u$}_{\mbox{}_{\scriptstyle \!\:\!\delta}}
\!\:\!(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}^{\:\!2}
\!\;\!ds
$\
podem ser estimadas independentemente de $ \:\!\delta > 0 $. Isso será usado no Teorema 3.1 (Seção 