I'm studying the very basics of kinetic theory in Vlasov Poisson Systems, and the first equation I'm studying is the free transport equation, i.e.: $$\frac{\partial f}{\partial t}+v\cdot\nabla_{x}f=0$$$$f(0,x,v)=f^{0}(x,v)$$
where $\,f(t,x,v)\geq 0\,$ is the distribution function of particles. One of its properties is that we can define a conservated quantity called the kinetic energy: $$E_{kin}(t)=\frac{1}{2}\int_{\mathbb{R}^{3}_{x}}\int_{\mathbb{R}^{3}_{v}}|v|^{2}f(t,x,v)\,dx\,dv.$$
Now, I know that this is a conservated quantity (i.e. $\frac{d}{dt}E_{kin}=0$) but I want to prove it, at least formally. I started multiplying the free transport equation by $\frac{|v|^{2}}{2}$ and then integrating everything with respect of x and v, and perhaps this is a dumb step but I can´t get rid of the $\int_{R^{3}_{x}}\int_{R^{3}_{v}}\frac{|v|^{2}}{2}v\cdot\nabla_{x}f\,dx\,dv$ term, how can I see that this term is zero? Any help is welcome, regards.
Usual steps: (1) differentiate energy with respect to time; (2) use the PDE. $$\frac{1}{2}\int_{\mathbb{R}^{3}_{x}}\int_{\mathbb{R}^{3}_{v}}|v|^{2}\frac{\partial f}{\partial t}(t,x,v)\,dx\,dv = - \frac{1}{2}\int_{\mathbb{R}^{3}_{x}}\int_{\mathbb{R}^{3}_{v}}|v|^{2}v\cdot \nabla_x f(t,x,v)\,dx\,dv $$ Often there is step (3): integrate by parts. It's not really needed here, though. For each fixed $v$, we have $$\int_{\mathbb{R}^{3}_{v}} v\cdot \nabla_x f(t,x,v)\,dx = 0$$ because this is the directional derivative of $f$ in direction $v$, integrated over the entire space. We must assume, of course, that $f$ vanishes at infinity.
Solutions with mass going off into infinity or coming from infinity don't have to obey conservation laws since the system isn't really closed.