## 2 A Generalized Mathematical Model
### Reinforcing Non-Negative Species Distributions  
Assuming the system above has a unique solution, one obtains that the solution of the system (2a)-(2e) can be identified, for \(p_{S}=\
abla c_{S}\), with the solution of the following minimization problem: Find \((c_{S,LS},p_{S,LS})_{S\in\{A,B,C\}}\in X\), such that  
\[(c_{S,LS},p_{S,LS})_{S\in\{A,B,C\}}=\operatorname*{argmin}_{\left(p_{S},q_{S} \right)_{S\in\{A,B,C\}}\in X}LS\big{(}(v_{S},q_{S})_{S\in\{A,B,C\}}\big{)}, \tag{3}\]  
where \(X\) is a Banach space, which will be defined later, and the non-linear least-squares functional \(LS\big{(}(v_{S},q_{S})_{S\in\{A,B,C\}}\big{)}\) is given through the following identity:  
\[\begin{split}& LS\big{(}(v_{S},q_{S})_{S\in\{A,B,C\}}\big{)}:=\\ &\big{\|}D_{A}\mathrm{div}q_{A}+k_{2}v_{C}-k_{1}v_{A}v_{B}+f_{A} \big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}q_{A}-\
abla v_{A}\big{\|}_{L^{2}(\Omega) }^{2}\\ &\quad+\big{\|}D_{B}\mathrm{div}q_{B}+k_{2}v_{C}-k_{1}v_{A}v_{B}+f _{B}\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}q_{B}-\
abla v_{B}\big{\|}_{L^{2}( \Omega)}^{2}\\ &\quad+\big{\|}D_{C}\mathrm{div}q_{C}-k_{2}v_{C}+k_{1}v_{A}v_{B}+ f_{C}\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}q_{C}-\
abla v_{C}\big{\|}_{L^{2}( \Omega)}^{2}\\ &\quad+\sum_{S\in\{A,B,C\}}\left[\big{\|}v_{S}-g_{S,D}\big{\|}_{L ^{2}(\Omega)}^{2}+\big{\|}\
abla v_{S}\cdot v_{\Gamma_{N}}-g_{S,N}\big{\|}_{L^ {2}(\Gamma_{D})}^{2}\right]\end{split}\]  
For the last terms of the least squares functional above, one has to note that the space \(L^{\infty}(\Omega)\) is continuously embedded into the space \(L^{2}(\Omega)\) and, hence, the evaluation of the terms \(\big{\|}c_{S}-g_{S,D}\big{\|}_{L^{2}(\Gamma_{D})}^{2}\) and \(\big{\|}\
abla c_{S}\cdot v_{\Gamma_{N}}-g_{S,N}\big{\|}_{L^{2}(\Gamma_{D})}^ {2}\) are formally correct and simplify the calculations needed to implement a corresponding solver.  
In the following, the space \(X\) has to be fixed. The space \(X\) is defined s.t. the non-linear functional \(LS\big{(}(v_{S},q_{S})_{S\in\{A,B,C\}}\big{)}\) is well defined, for each element \((v_{S},q_{S})_{S\in\{A,B,C\}}\in X\).  
First, note that, for each \(S\in\{A,B,C\}\), the conditions \(\mathrm{div}q_{S}\in L^{2}(\Omega)\) and \(q\in L^{2}(\Omega;\mathbb{R}^{d})\) have to be fulfilled, since for \(d=1\), the divergence reduces to a simple derivative, i.e., \(\mathrm{div}q_{S}=\frac{q}{2\pi}q_{S}\), one obtains, for \(d=1\), the inclusion \(q_{S}\in H^{1}(\Omega)\) and for \(d=2,3\) the inclusion \(q_{S}\in H(\mathrm{div};\Omega)\), for the definition of \(H(\mathrm{div};\Omega)\) see [36; 40].  
Furthermore, one observes that the inclusion \(v_{A}v_{B},v_{C}\in L^{2}(\Omega)\) has to be fulfilled, a sufficient condition to fulfil the inclusion above is \(v_{A},v_{B}\in L^{4}(\Omega)\) and \(\
abla v_{A},\
abla v_{B},\
abla v_{C}\in L^{2}(\Omega;\mathbb{R}^{d})\), whence \(v_{A},v_{B}\in W^{1A}(\Omega)\) and \(v_{C}\in H^{1}(\Omega)\). The spaces \(W^{k,p}(\Omega)\) and \(H^{k}(\Omega)\) denote the usual and well known Sobolev spaces, cf. [36; 40]. This gives, for \(d=1\), the identity \(\big{(}W^{1A}(\Omega)\times H^{1}(\Omega)\big{)}^{X^{2}}\times\big{(}H^{1}( \Omega)\times H^{1}(\Omega)\big{)}\) and for \(d=2\) the identity \(\big{(}W^{1A}(\Omega)\times H(\mathrm{div},\Omega)\big{)}^{X^{2}}\times\big{(}H ^{1}(\Omega)\times H(\mathrm{div},\Omega)\big{)}\).  
The reformulation discussed in this article is to introduce the positivity of \(c_{A}\), \(c_{B}\), \(c_{C}\) as side condition, cf. [11; 41].  
\[\begin{split}\text{Find }\big{(}c_{S,LS},p_{S,LS}\big{)}_{S\in\{A,B,C\}} \in X,\text{ such that}\\ \big{(}c_{S,LS},p_{S,LS}\big{)}_{S\in\{A,B,C\}}& =\operatorname*{argmin}_{\left(v_{S},q_{S}\right)_{S\in\{A,B,C\}}\in X}& LS \big{(}(v_{S},q_{S})_{S\in\{A,B,C\}}\big{)}\\ &\quad\text{\bf s.t.}&\quad 0\leq v_{S},\quad\forall S\in\{A,B,C \}\end{split} \tag{4}\]where the non-linear least-squares functional \(LS\big{(}(v_{S},q_{S})_{S\in\{A,B,C\}}\big{)}\) is the same as in (3).