Lions and Paul claim in their 1993 paper "Sur les mesures de Wigner" that the Hamiltonian flow


$$\dot{x} = \xi, \quad \dot{\xi} = - \nabla V(x) $$


of the Vlasov equation


$$\partial\_t f + \xi \cdot \nabla\_x f + \nabla\_x V \cdot \nabla\_{\xi} f = 0$$


is well defined if $V\in C^{1,1}(\mathbb{R}^d)$ and $\exists C>0$ such that


$$ V(x) \geq -C(1+|x|^2)$$


(Théorème IV.1)


Neither do they prove it nor do they give a reference for this claim. It is, however, not obvious to me.



Why is this Hamiltonian flow of the Vlasov equation well defined?
I have found an answer.


The condition $V\in C^{1,1}$ ensures that $\nabla V$ is Lipschitz which implies the existence of a global solution in the neighborhood of every point for the Cauchy problem of the differential system $\dot{x} = \xi, \dot{\xi} = -\nabla V(x)$.


We have the following inequality for the energy $E(t)$


$$E(t) := \frac12 |\xi(t)|^2 + V(q(t)) > \frac12 |\xi(t)|^2 -C(1+|x(t)|^2) \\ > \frac12 |\xi(t)|^2 - 2C - 2C|x(t)|^2 $$


Therefore,


$$ |\xi(t)|^2 < 2E +2C + 2|x(t)|^2.$$


Moreover,


$$\frac{d}{dt} |x(t)|^2 = 2\xi(t) x(t) \leq |\xi(t)|^2 +|x(t)|^2 \leq 2E+2C +(2C+1)|x(t)|^2.$$


Therefore, $|\xi(t)|+|x(t)|$ are exponentially bounded by Gronwall's inequality and hence do not blow up in finite time. Then we can conclude that the flow exists globally.


