)
} $ é o divergente (espacial) de $ \:\!\mbox{\boldmath $u$}(\cdot,t) $; analogamente, $ {\displaystyle
\mbox{\boldmath $u$} \cdot \nabla \mbox{\boldmath $u$}
\;\!=\;\!
u_{\mbox{}_{\!\:\!1}} D_{\mbox{}_{\!\:\!1}} \mbox{\boldmath $u$}
+ ... +
u_{\mbox{}_{\scriptstyle \!\:\!n}}
D_{\mbox{}_{\scriptstyle \!\:\!n}} \mbox{\boldmath $u$}
} $, etc. $ {\displaystyle
\| \;\!\cdot\;\!
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{n})}}
\!\;\!
} $, $ 1 \leq q \leq \infty $, denota a norma tradicional do espaço de Lebesgue $ L^{q}(\mathbb{R}^{n}) $, pondo-se, para $ 1 \leq q < \infty $:

\
$$\tag{1.19$a$}
\|\, \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{n})}}
\;\!=\;
\Bigl\{\,
\sum_{i\,=\,1}^{n} \int_{\mathbb{R}^{n}} \!
|\:u_{i}(x,t)\,|^{q} \;\!dx
\,\Bigr\}^{\!\!\:\!1/q}$$\
$$\tag{1.19$b$}
\|\, D \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{n})}}
\;\!=\;
\Bigl\{\,
\sum_{i, \,j \,=\,1}^{n} \int_{\mathbb{R}^{n}} \!
|\, D_{\!\;\!j} \;\!u_{i}(x,t)\,|^{q} \;\!dx
\,\Bigr\}^{\!\!\:\!1/q}$$\
$$\tag{1.19$c$}
\|\, D^{2} \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{n})}}
\;\!=\;
\Bigl\{\!\!
\sum_{\mbox{} \;\;i, \,j, \,\ell \,=\,1}^{n}
\!\;\! \int_{\mathbb{R}^{n}} \!
|\, D_{\!\;\!j} \:\!D_{\ell} \, u_{i}(x,t)\,|^{q} \;\!dx
\,\Bigr\}^{\!\!\:\!1/q}$$\
e, mais geralmente, para $ m \geq 1 $ qualquer:\
\
$$\tag{1.19$d$}
\|\, D^m \:\! \mbox{\boldmath $u$}(\cdot, t)\,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{n})}}
=
\biggl(\;\! \sum_{i\,=\,1}^{n}
\sum_{\;\!j_{\mbox{}_{1}}=\,1}^{n}
\!\cdots\!
\sum_{j_{\mbox{}_{m}}=\,1}^{n}
\int_{\mbox{}_{\mbox{\scriptsize $\!\;\!\mathbb{R}^{n}$}}}
\!\!\!\!\;\!
|\;\!D_{\scriptstyle \!\;\!j_{\mbox{}_{1}}}
\!\!\:\!\cdots
D_{\scriptstyle \!\;\!j_{\mbox{}_{m}}}
u_{i}(x,t)\:|^q\, dx\:\!\biggr)^{\!\!\:\!1/q}\!\!\!\!,$$\
denotando-se por $ {\displaystyle
\;\!
\|\, \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R}^{n})}}
\!=\;\!
\max \, \bigl\{\,
\|\,u_{i}(\cdot,t)\,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R}^{n})}}
\!\!: \, 1 \leq i \leq n
\,\bigr\}
} $ o 