Let
$$\begin{aligned} \varvec{\varphi }= \{ \varphi ^1, \ldots , \varphi ^n\} \end{aligned}$$
(8)
be a list of species concentrations and assume that the function
\({\mathcal {F}}\) depends on
\(\varvec{\varphi }\) and
\({\mathcal {H}}\) on
\(\mathrm{grad}\varvec{\varphi }\) such that
$$\begin{aligned} {\mathcal {F}}(\varvec{\varphi }) = {\mathcal {F}}(\varphi ^1,\ldots ,\varphi ^n)\quad \text {and}\quad {\mathcal {H}}(\mathrm{grad}\varphi ^{1},\ldots ,\mathrm{grad}\varphi ^\alpha +\epsilon {\varvec{o}},\ldots ,\mathrm{grad}\varphi ^\varsigma -\epsilon {\varvec{o}},\ldots ,\mathrm{grad}\varphi ^{n}). \end{aligned}$$
(9)
Constraint (
5), with (
4), implies that the set of concentrations
\(\varvec{\varphi }\) must be
\(0< \varphi ^\alpha < 1\). If we vary one concentration
\(\varphi ^\alpha \) while holding all others fixed violates constraint (
5). Thus, the conventional partial derivative on functions such as
\({\mathcal {F}}\) and
\({\mathcal {H}}\), on which constraint (
5) is active, is not appropriately defined. To overcome this shortcoming, Larché and Cahn [
9] defined the following operation
$$\begin{aligned} \frac{\partial ^{(\sigma )} {\mathcal {F}}(\varvec{\varphi })}{\partial \varphi ^\alpha } = \frac{\text {d}}{\text {d} \epsilon } {\mathcal {F}}(\varphi ^{1},\ldots ,\varphi ^\alpha +\epsilon ,\ldots ,\varphi ^\sigma -\epsilon ,\ldots ,\varphi ^{n}) \Bigr |_{\epsilon =0} \end{aligned}$$
(10)
and we extended it to
$$\begin{aligned} \dfrac{\partial ^{(\sigma )} {\mathcal {H}}(\mathrm{grad}\varvec{\varphi })}{\partial (\mathrm{grad}\varphi ^\alpha )} = \dfrac{\text {d}}{\text {d}\epsilon } {\mathcal {H}}(\mathrm{grad}\varphi ^{1},\ldots ,\mathrm{grad}\varphi ^\alpha +\epsilon {\varvec{o}},\ldots ,\mathrm{grad}\varphi ^\sigma -\epsilon {\varvec{o}},\ldots ,\mathrm{grad}\varphi ^{n}) \Bigr |_{\epsilon =0}, \end{aligned}$$
(11)
where
\({\varvec{o}}\) is a vector fully populated with ones, and in which we choose any two concentrations
\(\varphi ^\alpha \) and
\(\varphi ^\sigma \) from the set of variables. Then, we introduce an infinitesimal change
\(\epsilon \) in
\(\varphi ^\alpha \), which induces the opposite infinitesimal variation
\(\epsilon \) onto
\(\varphi ^\sigma \), while holding all other variables unchanged. Thus, this definition satisfies (
5) by construction while we express the concentration
\(\varphi ^\sigma \) as
$$\begin{aligned} \varphi ^\sigma = 1 - \sum _{\begin{array}{c} \alpha =1 \\ \alpha \ne \sigma \end{array}} ^{n} \varphi ^\alpha . \end{aligned}$$
(12)
In multicomponent Cahn–Hilliard systems, we incorporate cross-diffusion gradient energy coefficients
\(\Gamma ^{\alpha \beta }\) into the free-energy definition and obtain the following free-energy density
$$\begin{aligned} {\hat{\psi }}(\varvec{\varphi },\mathrm{grad}\varvec{\varphi }) {:}= f(\varvec{\varphi }) + \sum _{\alpha =1}^n\sum _{\beta =1}^n\Gamma ^{\alpha \beta }\mathrm{grad}\varphi ^\alpha \cdot \mathrm{grad}\varphi ^\beta . \end{aligned}$$
(13)
Elliott and Garcke in [
10] prove that multicomponent systems are well posed when
\(\Gamma ^{\alpha \beta }\) is positive definite, among other conditions. We show that this condition is sufficient but not necessary. To do so, we extend the ideas of Larché–Cahn and define a
constrained inner product on a constrained space. We consider a set of vectors
\(\{{\varvec{p}}^\alpha \}_{\alpha =1}^n\) subject to the following constraint
$$\begin{aligned} \sum _{\alpha =1}^n{\varvec{p}}^\alpha = \mathbf{0}, \end{aligned}$$
(14)
and use the following inner product
$$\begin{aligned} \sum _{\alpha =1}^n\sum _{\beta =1}^n\Gamma ^{\alpha \beta }{\varvec{p}}^\alpha \cdot {\varvec{p}}^\beta . \end{aligned}$$
(15)
Let each entry of
\(\Lambda ^{\alpha \beta }\) be a single number
\(\kappa \). Thus, due to (
14),
\(\{{\varvec{p}}^\alpha \}_{\alpha =1}^n\) is in the null space of
\(\Lambda ^{\alpha \beta }\), that is,
\(\text {Null}(\Lambda ^{\alpha \beta })=\{{\varvec{p}}^\alpha \}_{\alpha =1}^n\). Similarly, if each row of
\(\Lambda ^{\alpha \beta }\) is given by the same entry
\(\kappa ^\beta \), we arrive to the same conclusion. For any of these cases, we have that
$$\begin{aligned} \sum _{\alpha =1}^n\sum _{\beta =1}^n\Gamma ^{\alpha \beta }{\varvec{p}}^\alpha \cdot {\varvec{p}}^\beta =\sum _{\alpha =1}^n\sum _{\beta =1}^n(\Gamma ^{\alpha \beta }+\Lambda ^{\alpha \beta }){\varvec{p}}^\alpha \cdot {\varvec{p}}^\beta . \end{aligned}$$
(16)
We impose constraint (
14) with respect to the component
\(\sigma \) to the quadratic form (
15) to obtain
$$\begin{aligned} \sum _{\alpha =1}^n\sum _{\beta =1}^n\Gamma ^{\alpha \beta }{\varvec{p}}^\alpha \cdot {\varvec{p}}^\beta =\sum _{\begin{array}{c} \alpha =1 \\ \alpha \ne \sigma \end{array}}^{n}\sum _{\begin{array}{c} \beta =1 \\ \beta \ne \sigma \end{array}}^{n}(\underbrace{\Gamma ^{\alpha \beta }+\Gamma ^{\sigma \sigma }-\Gamma ^{\alpha \sigma }-\Gamma ^{\sigma \beta }}_{\Gamma ^{\alpha \beta }_\sigma }{\varvec{p}}^\alpha \cdot {\varvec{p}}^\beta . \end{aligned}$$
(17)
We reinterpret this result as an inner product in an unconstrained space of dimension
\(n-1\) with a
non-invertible mapping
\(\Gamma ^{\alpha \beta }\mapsto \Gamma ^{\alpha \beta }_{\sigma }\) defined as
$$\begin{aligned} \Gamma ^{\alpha \beta }_{\sigma }{:}=\Gamma ^{\alpha \beta }+\Gamma ^{\sigma \sigma }-\Gamma ^{\alpha \sigma }-\Gamma ^{\sigma \beta }. \end{aligned}$$
(18)
Consequently, the problem is well posed if
\(\Gamma ^{\alpha \beta }_{\sigma }\) is positive definite. Moreover,
\(\Gamma ^{\alpha \beta }\) can be indefinite without compromising the well posedness of the problem. Now, let
\(\Gamma ^{\alpha \beta }\) be a diagonal matrix such that
$$\begin{aligned} \Gamma ^{\alpha \beta }=\kappa \,\delta ^{\alpha \beta }. \end{aligned}$$
(19)
From (
16), we rewrite
\(\Gamma ^{\alpha \beta }\) as
$$\begin{aligned} \Gamma ^{\alpha \beta }=-\kappa \left( 1^{\alpha \beta }-\delta ^{\alpha \beta }\right) , \end{aligned}$$
(20)
where
\(1^{\alpha \beta }\) is a constant matrix populated by ones and
\(\delta ^{\alpha \beta }\) is the Kronecker delta, both of dimension
n. Although matrix (
20) has a null diagonal, the mapping defined by (
18) is identical to the one of the diagonal matrices (
19) for all vectors that satisfy constraint (
14).