## 2 Mathematical Model and Analysis
### Existence of a Weak Solution
**_A function \(U^{n}(t,x)=\sum_{i=1}^{n}\ b_{i}^{n}(t)w^{i}(x)\) is known as the \(n\)-th Galerkin approximation, if_  
\[\int_{\Omega}\frac{\partial U^{n}}{\partial t}.\ w^{j}\ dx+\int_{\Omega}\ (U^{n}.\
abla U^{n}).\ w^{j}\ dx+\theta\int_{\Omega}\ \
abla U^{n}:\
abla w^{j}\ dx=0,\text{ for all }j=1(1)n \tag{4}\]  
\[U^{n}(0,x)=\sum_{i=1}^{n}a_{i}w^{i}(x), \tag{5}\]  
_where \(a_{i}=\int_{\Omega}\ u_{0}(x).\ w^{i}(x)\ dx\)._  
_To a system of ordinary differential equations for \(\left\{b_{i}^{n}(t)\right\}_{i=1}^{n}\), then the above equation can be re-written as_  
\[b_{j}^{n}(t)+b_{k}^{n}(t)b_{l}^{n}(t)\int_{\Omega}\ \Big{(}w^{k}.\
abla w^{l} \Big{)}.\ w^{j}dx+\vartheta\lambda_{j}b_{j}^{n}(t)=0.j=1(1)n, \tag{6}\]  
_as \(\int_{\Omega}w^{i}.\ w^{j}dx=\delta_{ij}\) with \(b_{j}^{n}(0)=a_{j}\)._  
_The above equation represents a system of ordinary differential equations._  
Step 2: Existence of a Unique Generalized Solution  
To the system of ordinary differential equations (ODE), we shall apply the Caratheodory theory, which entails the existence of a unique generalized solution \(b_{j}(t)\in C[0,t)\). Multiplying the system of ODE with \(b_{j}(t)\) and summing up over \(j=1(1)n\). Integrating over the open interval \((0,t)\) we obtain  
\[\int_{0}^{t}\ \frac{1}{2}\sum_{j=1}^{n}\ \frac{d}{dt}\Big{|}b_{j}\Big{|}^{2}d \tau+\int_{0}^{t}\ b_{k}b_{l}b_{j}\ \int_{\Omega}\ (w^{k}.\
abla w^{l}).w^{j}dx\ d\tau+\theta\int_{0}^{t}\sum_{j=1}^{n}\ \frac{d}{dt}\Big{|}b_{j}\Big{|}^{2} \lambda_{j}d\tau=0\]  
\[\Rightarrow\int_{0}^{t}\ \frac{1}{2}\frac{d}{dt}\|U^{n}(t)\|_{2}^{2}\ d\tau+\int_{0}^ {t}\ \int_{\Omega}\ (U^{n}.\
abla U^{n}).U^{n}dx\ d\tau+\theta\int_{0}^{t}\ \int_{\Omega}\ |\
abla U^{n}|^{2}\ dx\ d\tau=0\]\(\Rightarrow\frac{1}{2}\parallel U^{n}(t)\|_{2}^{2}+\theta\int_{0}^{t}\ \ \|\
abla U^{n}\|_{2}^{2}\ d\tau\ \leq\frac{1}{2}\|U^{n}(0)\|_{2}^{2}=\frac{1}{2}\sum\limits_{j=1}^{n}\ a_{j}^{ 2}\leq\|u_{0}(x)\|_{2}^{2}\leq C(u_{0})\)  
\(\Rightarrow b_{j}(. )\) are bound in time, \(j=1(1)n\)  
This implies that the sequence \(\{U^{n}\}\) is bounded in spaces \(L^{\infty}(0,t;\left(L^{2}\ (\Omega)\right)^{N})\) and \(L^{2}(0,t;\left(W^{1,2}\ (\Omega)\right)^{N})\) uniformly with respect to \(n\). Now, let \(\phi\in L^{2}(0,t;\left(W_{0,\ div}^{1,2}\ (\Omega)\right)^{N})\). Thus, \(\phi(t,x)=\sum_{k=1}^{\infty}\ a_{k}(t)w^{k}(x)\), \(a_{k}(t)=\int_{\Omega}\ \phi(t,x).w^{k}(x)\ dx\)  
Denoting \(\phi^{n}(t,x)=\sum_{k=1}^{n}\ a_{k}(t)w^{k}(x)\)  
Hence, \(\|\phi^{n}\|_{L^{2}(0,t;\left(W_{0,\ div}^{1,2}\ (\Omega)\right)^{N})}\leq\|\ \phi\|_{L^{2}(0,t;\left(W_{0,\ div}^{1,2}\ (\Omega)\right))}\)  
\(\Rightarrow\|\frac{\partial U^{n}}{\partial t}\|_{L^{2}(0,t;\left(W_{0,\ div}^{1,2} (\Omega)\right)^{*})}=Sup_{\phi\in L^{2}(0,t;\left(W_{0,\ div}^{1,2}(\Omega) \right))\ \|\phi\|_{2}\leq\big{|}\int_{0}^{t}\ \int_{\Omega}\ \frac{\partial U^{n}}{ \partial t}\phi\ dx\ dt\big{|}\)  
\(=Sup\Big{|}\int_{0}^{t}\ \int_{\Omega}\ (U^{n}.\
abla U^{n}).\phi^{n}\ dx\ dt-\theta \int_{\Omega}\ \
abla U^{n}:\
abla\phi^{n}\ dx\ dt\Big{|}\)  
\(\leq Sup\Big{[}\theta\|\
abla U^{n}\|_{L^{2}(0,t;\left(L^{2}(\Omega)\right)^{ N})}\Big{]}\|\phi^{n}\|_{L^{2}(0,t;\left(W^{1,2}(\Omega)\right)^{N})}+C\|\
abla U^{n}\|_{L^{ \infty}(0,t;\left(L^{2}(\Omega)\right)^{2})}\|\
abla U^{n}\|_{L^{2}(0,t;\left( L^{2}(\Omega)\right)^{4})}\|\phi\
abla\phi^{n}\|_{L^{2}(0,t;\left(L^{2}(\Omega) \right)^{4})}\)  
And hence we get  
\(\|\frac{\partial U^{n}}{\partial t}\|_{L^{2}(0,t;\left(W_{0,\ div}^{1,2}( \Omega)\right)^{*})}\leq C(u_{0})\)  
Considering the equality  
\(\frac{1}{2}\ \|U^{n}(t)\|_{2}^{2}+\theta\int_{0}^{t}\ \int_{\Omega}\ | \
abla U^{n}|^{2}\ dx\ d\tau-\frac{1}{2}\|U^{n}(0)\|_{2}^{2}=0\)  
Multiplying by \(\psi\in C_{0}^{\infty}(0,t)\), \(\psi\geq 0\) on \([0,t]\) and integrate over \([0,T]\). We have  
\(\int_{0}^{T}\ \bigg{[}\frac{1}{2}\ \|U^{n}(t)\|_{2}^{2}\psi+\theta\int_{0}^{t}\ \int_{\Omega}\ | \
abla U^{n}|^{2}\ dx\ d\tau\ \psi-\frac{1}{2}\|U^{n}(0)\|_{2}^{2}\ \psi\bigg{]}dt=0\)  
Letting \(n\rightarrow\infty\).