# Is this an elements of the Sobolev-Space $W^{1,2}\left(0,T,H^1(\Omega),H^1(\Omega)'\right)$?

Definition
Let $p\in \mathbb{Z}$, $a<b \in \mathbb{R}$, and $X$, $Y$ be real Hilbert spaces. We define $$W^{1,p}(a,b,X,Y) := \left\{ \varphi \, \Big| \, \varphi \in L^p(a,b,X),\, \varphi' \in L^p(a,b,Y) \right\}.$$

I am working with the Bidomain-Model which, over a time intervall [0,T], describes the electrical behaviour of the myocardial muscle considered as $\Omega \subset \mathbb{R}^3$.
This model has partial differential equations which involve the intra- and extracellular voltages, $u$ and $v$, which are functions of this type: $$u=u(t,x):[0,T]\times\Omega\rightarrow\mathbb{R}$$ $$v=v(t,x):[0,T]\times\Omega\rightarrow\mathbb{R}$$

The model-equation which describes the relationship between these two is $$0 = A_iu-Cv$$ with the operators $A_i = -div(\sigma_i\nabla \cdot)$ and $C = div\left((\sigma_i + \sigma_e)\nabla \cdot \right)$, for my pourposes.
As most of us know, if we use a computer to simulate the model, we would get approximations $\widetilde{u}$ and $\widetilde{v}$ and thus the norm $$\|A_i\widetilde{u}-C\widetilde{v} \|$$
becomes relevant.
In order to analytically represent and bound this residual, I am still missing to see in which space the functions $$\text{res}(t,x):=A_i\widetilde{u}(t,x)-C\widetilde{v}(t,x) \qquad \text{and}$$ $$\dot{\text{res}},$$ its time derivative, live and which norms this spaces have.

Is it true that $$\text{res} : [0,T] \rightarrow H^1(\Omega) \quad \text{and}$$ $$\dot{\text{res}} : [0,T] \rightarrow \left[H^1(\Omega) \rightarrow \mathbb{R}\right]\text{?}$$ Does $$\text{res} \in L^{2} \left(0,T,H^1(\Omega)\right) \,\, \text{and}$$ $$\dot{\text{res}} \in L^{2} \left(0,T,H^1(\Omega)'\right)\text{?} \quad$$ Thus following that $$\text{res} \in W^{1,2}\left(0,T,H^1(\Omega),H^1(\Omega)'\right) \,\, \text{?}$$

## 1 Answer

It all seems to be right except for some fine details.
$\dot{\text{res}}$ can be interpreted in two ways:
First, as a function $$\dot{\text{res}} : [0,T] \rightarrow H^1(\Omega),$$ which maps time-points to (state)functions in $H^1(\Omega)$. Under this interpretation $$\dot{\text{res}} \in L^2\left(0,T,H^1(\Omega)\right).$$
Second, as a linear form $$\dot{\text{res}} : [0,T] \rightarrow H^1(\Omega)^*,$$ which maps time-points to rate-of-change-functions over (state)functions in $H^1(\Omega)$. This means that $\dot{\text{res}}$ is a linear form over $H^1(\Omega)$, thus $$\dot{\text{res}} \in L^2\left(0,T,H^1(\Omega)^*\right).$$ Forthermore you should have in mind that thanx to the Fréchet-Riesz representation theorem it holds that $$L^2\left(0,T,H^1(\Omega)\right) \cong L^2\left(0,T,H^1(\Omega)^*\right).$$ This spaces are isometrically isomorphic and this is what lies behind the fact, that $\dot{\text{res}}$ can be interpreted as a function and/or a linear form. Knowing this it is now clear that taking $\dot{\text{res}}$ as a linear form, then for the function of your interest, it holds that $$res := A_i \widetilde{u} - C\widetilde{v} \quad \in \quad W^{1,2}\left(0,T,H^1(\Omega),H^1(\Omega)^*\right).$$

  Please also consider this questions. And the comment about characterizing the norms thorough the Fourier-transform.