$$\begin{aligned}
\int_0^T \iint u(t,x,v)u_{-}(t,x,v) \,dv\,dx\,dt & =  -\int_{A^c}  {\ensuremath{\left|u_{-}(t,x,v)\right|}}^2 \,dv\,dx\,dt\\
& =  - \int_0^T\iint {\ensuremath{\left|u_{-}(t,x,v)\right|}}^2 \,dv\,dx\,dt.\end{aligned}$$ Hence, we find $$\lambda{\ensuremath{\left\langle  u,  u_- \right\rangle}} = -\lambda \Vert u \Vert_2$$ After a similar analysis for the other terms above, we find $${\ensuremath{\left\langle F \cdot \nabla_v  u,   u_- \right\rangle}} = -{\ensuremath{\left\langle F \cdot \nabla_v  u_-,   u_- \right\rangle}}.$$ For the diffusion term, we proceed similarly and integrate by parts to find $$\begin{aligned}
- {\ensuremath{\left\langle {\nabla}_v\cdot (D {\nabla}_v \cdot u),   u_- \right\rangle}}
&= {\ensuremath{\left\langle D {\nabla}_v u , {\nabla}_v u_{-} \right\rangle}}\\
&= -{\ensuremath{\left\langle D{\nabla}_v u_{-}, {\nabla}_v   u_- \right\rangle}}\\
&\leq  0,\end{aligned}$$ since $D$ is positive semi-definite. Therefore, using these identities with (\[Tartar\]) we have the inequality $${\ensuremath{\left\langle  f,   u_- \right\rangle}} \leq \frac{1}{2} \left ( \iint {\ensuremath{\left|  u_-(0,x,v)\right|}}^2 \,dx\,dv - \iint {\ensuremath{\left|  u_-(t,x,v)\right|}}^2 \,dx\,dv \right ) - {\ensuremath{\left\langle F \cdot \nabla_v  u_-,   u_- \right\rangle}}  - \lambda {\ensuremath{\left\|   u_- \right\|}}_2^2$$ By assumption, $ h_0(x,v) \geq 0$ and thus $  u_-(0,x,v) = 0$. The first term above is then nonpositive and $${\ensuremath{\left\langle  f,   u_- \right\rangle}}  \leq  - {\ensuremath{\left\langle F \cdot \nabla_v  u_-,   u_- \right\rangle}}  - \lambda {\ensuremath{\left\|   u_- \right\|}}_2^2.$$ Lastly, we integrate by parts to find $$\begin{aligned}
-{\ensuremath{\left\langle F \cdot \nabla_v  u_-,   u_- \right\rangle}} &=  -\int_0^T \iint F(t,x,v)\cdot \nabla_v 