I was trying to find the weak formulation of the following problem
\begin{align} -\vartheta\Delta u + b \cdot \nabla u &= f \quad \text{in $\Omega$} \\ u &= 0 \quad \text{in $\partial \Omega$} \end{align}
where $b:\bar{\Omega}\rightarrow \mathbb{R}^2, b\in C^1(\Omega)$ is a divergence free velocity field. I have computed the weak formulation by multiplying both sides with a test function and integrating over the domain. I have
\begin{equation} a(u,v) = \int_{\Omega} \vartheta \nabla u \cdot \nabla v+ (b \cdot \nabla u) v \end{equation}
Is this done? I don't know how to use the fact that $b$ is divergence free. Next I wanted to show that the bilinear form is bounded. Normally I would try and do this using the $L^2$ Norm and a Sobolev Inequality $\|u\| \leq C\|Du\|$. But I feel I am doing something wrong because it is not working.
Your blf as in the comment, i.e.,
$$a(u,v) = \int_\Omega \vartheta \nabla u \cdot \nabla v -\int_\Omega (bu) \cdot \nabla v $$
for all $v \in H_0^1(\Omega)$ is correct. It is bounded in $H_0^1 \times H_0^1$ since
$$\begin{aligned}|a(u,v)| &\leq \vartheta \|\nabla u\|_{L^2} \|\nabla v\|_{L^2} + \|b\|_{L^\infty} \|u\|_{L^2(\Omega)} \|\nabla v\|_{L^2(\Omega)} \\ &\leq (\vartheta+\|b\|_{L^\infty}) \|u\|_{H^1} \|v\|_{H^1}. \end{aligned}$$