Navier Stoke's system

Question

Lemma: Let $\bar{g}\in\left(H^{\frac{1}{2}}(\Gamma)\right)^2$ such that $\int\limits_\Gamma \bar{g}.v=0$. Then $\exists$ $\bar u\in (H^1(\Omega))^2$ such that div($\bar u$)=$0$, $\bar u=\bar g$ on $\Gamma.$ [where $\Omega$ is bounded connected set with lipschitz boundary $\Gamma$].