## 2 Mathematical Model and Analysis
### Existence of a Weak Solution  
Weak formulation of the governing equations mentioned above can be obtained by multiplying the Navier-Stokes momentum equation with the test function \(\phi\) taken from the test space \(\left(C_{0}^{\infty}(\Omega)\right)^{N}\) with vanishing divergence, integrating over the domain of interest. Applying the Gauss formula as a subsequent step, we obtain  
\[\int\limits_{\Omega}\frac{\partial u}{\partial t}.\phi\ dx+\int \limits_{\Omega}\left(u.\
abla u\right).\ \phi\ dx+\theta\int\limits_{\Omega}\
abla u:\
abla\phi\ dx+\int \limits_{\Omega}\ p.\ \
abla\phi\ dx=0 \tag{1}\]  
**Definition 1. **_Let \(\Omega\) be a subspace of \(R^{2}\). Let \(u_{0}\in L^{2}_{0,div}(\Omega)\). Then the function \(u\in L^{2}\Big{(}0,t;W^{1,2}_{0,div}(\Omega)\Big{)}\)\(\cap\)\(L^{\infty}\Big{(}0,t;L^{2}_{0,div}(\Omega)\Big{)}\) with \(\frac{\partial u}{\partial t}\) belonging to the dual space \(L^{1}(0,t;(W^{1,2}_{0,div}(\Omega))^{*})\) is said to be a weak solution or a variational solution of Navier-Stokes equation corresponding to the initial value of \(u\) considered, if \((\frac{\partial u}{\partial t},\phi)(W^{1,2}_{0,div}(\Omega))^{*}_{\cdot}(W^{ 1,2}_{0,div}(\Omega))\)\(+\int\limits_{\Omega}(u.\
abla u).\ \phi\ dx+\theta\int\limits_{\Omega}\
abla u:\
abla\phi\ dx=0\), for all test functions, picked from the space \(\phi\)\(W^{1,2}_{0,div}(\Omega)\) and almost all values of time from the open interval \((0,t)\), \(\int\limits_{\Omega}\ u(t,. )\ \phi\ dx=\int\limits_{\Omega}\ u_{0}.\ \phi\ dx,\ \forall\phi\in L^{2}_{0,div}(\Omega)\)._  
To achieve a sufficiently smooth solution to the momentum equations, we replace \(\phi\) by \(u\) to obtain,  
\[\int\limits_{\Omega}\frac{\partial u}{\partial t}.\phi\ dx+\int \limits_{\Omega}\left(u.\
abla u\right).\ \phi\ dx+\theta\int\limits_{\Omega}\
abla u:\
abla\phi\ dx=0 \tag{2}\]  
where the pressure term can be reconstructed by the principles considered by [PERSON] et al. [41]. \[\Rightarrow\frac{1}{2}\frac{d}{dt}\|u\|_{2}^{2}+\frac{1}{2}\int \limits_{\Omega}\left(u.\
abla|u|^{2}\right)dx+2\theta\int\limits_{\Omega}| \
abla u|^{2}dx=\int\limits_{\Omega}|u_{0}|^{2}dx\]\[\Rightarrow\frac{1}{2}\frac{d}{dt}\|u\|_{2}^{2}-\frac{1}{2}\int_{\Omega}\ \
abla.u\ |\
abla u|^{2}\ dx+\frac{1}{2}\int_{\partial\Omega}\ u.n\ |u|^{2}\ dS+2\theta\int_{\Omega}\ |\
abla u|^{2}dx=\int_{\Omega}\ |u_{0}|^{2}dx\]  
The second and third terms of the left-hand side vanish. Integrating the resultant expression concerning time variable \(t\), we get  
\[\int\limits_{\Omega}(|u(t)|^{2})dx+2\theta\int_{0}^{t}\ \int_{\Omega}\ |\
abla u|^{2}dx\ d \tau=\int_{\Omega}\ |u_{0}|^{2}dx \tag{3}\]  
This depicts the Energy equality for the problem under consideration. For a three-dimensional system, the above equality gains a weak form (energy inequality). **Theorem 1. **_Let \(\Omega\subset R^{2}\) be a bounded domain and \(u_{0}\) satisfies the assumptions made in the Definition 1 and \(0<t<\infty\). Then there exists one weak solution to the Navier-Stokes equation, that satisfies the initial condition in the sense \(\lim\limits_{t\to 0^{+}}\|u(t)-u_{0}\|_{2}=0\). Furthermore, \(u\in C([0,t];L^{2}_{0,div}(\Omega))\) and since \(u\) also belongs to \(C([0,t];(L^{2}(\Omega))^{2})\) and it satisfies energy equality._  
**Proof. ** The theorem is proved in 4 steps. \(\square\)  
Step 1: Galerkin Approximation or Formulation  
Let us consider the set \(\left\{w^{i}\right\}_{i=0}^{\infty}\). This is an orthogonal basis of \(W^{1,2}_{0,\ div}(\Omega)\) formed by the eigenfunctions of the Stokes operator. **Definition 2.