In the study of Navier Stokes Equations,
Let us consider the set:
$V: =${$ϕ∈C_c^{\infty}(Ω)$, div$ϕ =0$}.
$C_c^{\infty}(Ω)$:= space of $C^{\infty}(Ω)$ with compact support, where $Ω \subset \mathbb{R^n}$ is open.
We can then define the following fundamental space:
$H$: as the closure of $V$ in $L^2(Ω)$.
As in the reference, this space is of course Hilbert space equipped with the scalar product induced by that of $L^2(Ω)$.
Furthermore, and after some proofs,
$H=${$u∈L^2(Ω)$, div$u$ =0,$\frac{\partial u}{\partial n}:=\nabla u.n=0$}, where $n$ is an outward unit vector.
My Question is,
How to express $||u||_H$ where $u \in H$?
I am little lost because (as told in reference), $\exists$ a continuous embedding from $H$ to $L^2(Ω)$, so one can say
$||u||_{L^2(Ω)} \le$ cst.$||u||_H$
and if you answered me that $||u||_H:=||u||_{L^2(Ω)}$, $\forall u \in H$, I would be somehow confused, since then what is the benefit of telling that $H$ is continuously embedded in $L^2(Ω)$
Some help please!!