Bounding $\int_0^T\int_\Omega v|\nabla u|^2$ given that $v \in L^2(0,T;L^2(\Omega))$ and $u \in L^2(0,T;H^1(\Omega)) \cap L^\infty(0,T;L^2(\Omega))$?

Question

If $v \in L^2(0,T;L^2(\Omega))$ and $$u \in L^2(0,T;H^1(\Omega)) \cap L^\infty(0,T;L^2(\Omega)),\tag{1}$$ is possible to get a bound on the integral $$\int_0^T\int_\Omega v|\nabla u|^2$$ of the form $$\int_0^T\int_\Omega v|\nabla u|^2 \leq C\lVert v \rVert_{L^2(0,T;L^2(\Omega))}\times(\text{some norm of $u$ or $\nabla u$ that is finite (given that $u$ satisfies (1))})$$ (Notice the multiplcaton on the second line above)?

Young's/Holder's inequality are no use here I don't think.