I had solved a previous problem on $H^1_0(0,1)$ using the identity $$ \int_0^1 b(x)u'(x) u(x) = -\frac{1}{2} \int_0^1 b'(x) u(x)^2. $$
Is there any analogue to this for nice domains $\Omega$ where we have $$ \int_\Omega [\overset{\to}{b}(x)\cdot\nabla u(x)] u(x) = -\frac{1}{2}\int_\Omega \nabla \cdot \overset{\to}{b}(x) u(x)^2. $$ I'm rusty on vector calculus, but I think I need the divergence theorem somewhere.