$H_0^1(\Omega)$ in the study of the Navier-Stokes Equations

Question

This is cross-posted on MSE: https://math.stackexchange.com/q/1584519/9464

Let $\mathcal{V}$ be the space (without topology)

$$\displaystyle \mathcal{V}=\{u\in C_0^\infty(\Omega)\mid \nabla\cdot u=0\}$$

where $\Omega$ is a nonempty open connected subset of $\mathbb{R}^n$.

It is said in the Navier-Stokes Equations by Temam that the closure of $\mathcal{V}$ in $L^2(\Omega)$ and in $H_0^1(\Omega)$ (which is defined as the closure of $C_0^\infty(\Omega)$ in the Sobolev space $W^{1,2}(\Omega)$) are two basic spaces in the study of the Navier-Stokes equations. While it is quite clear what the closure of $\mathcal{V}$ in $L^2(\Omega)$ means, I don’t quite understand later one.

Isn't the closure of $\mathcal{V}$ in $H_0^1(\Omega)$ the same as the closure of $\mathcal{V}$ in $H^1(\Omega)$? Why bother mentioning the space $H_0^1(\Omega)$?

Answer 1

Yes, it is the same. I think that Roger (my advisor) wanted only to emphazise that because $\cal V$ is included in $H^1_0$, and the latter is a closed subspace of $H^1$, the closure of $\cal V$ is contained in $H^1_0$, hence its elements satisfy the boundary condition $u=0$.