Let $\bf{u}$ be a smooth complex vectorfield defined on the closed unit ball $B_1(0)\subset \mathbb{R}^3$. Let $\phi$ be any smooth complex function defined on $B_1(0)$.

My question is, is the following true:

$\textbf{Claim}:$ There exists a number $c>0$ and a function $g$ on $B_1(0)$ satisfying $\nabla g = {\bf{u}} g$ on $B_1(0)$ such that \begin{equation} \int_{B_1(0)} \left|\nabla \phi- {\bf{u}} \phi\right|^2 \geq c \int_{B_1(0)} \left|\phi-g\right|^2. \end{equation}

I suspect that the above claim is true but I have no proof at the moment.